| Step |
Hyp |
Ref |
Expression |
| 1 |
|
addsdilem.1 |
⊢ ( 𝜑 → 𝐴 ∈ No ) |
| 2 |
|
addsdilem.2 |
⊢ ( 𝜑 → 𝐵 ∈ No ) |
| 3 |
|
addsdilem.3 |
⊢ ( 𝜑 → 𝐶 ∈ No ) |
| 4 |
1 2
|
mulcut2 |
⊢ ( 𝜑 → ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) } ) <<s ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) } ) ) |
| 5 |
1 3
|
mulcut2 |
⊢ ( 𝜑 → ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) } ) <<s ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) } ) ) |
| 6 |
|
mulsval2 |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 ·s 𝐵 ) = ( ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) } ) |s ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) } ) ) ) |
| 7 |
1 2 6
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 ·s 𝐵 ) = ( ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) } ) |s ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) } ) ) ) |
| 8 |
|
mulsval2 |
⊢ ( ( 𝐴 ∈ No ∧ 𝐶 ∈ No ) → ( 𝐴 ·s 𝐶 ) = ( ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) } ) |s ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) } ) ) ) |
| 9 |
1 3 8
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 ·s 𝐶 ) = ( ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) } ) |s ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) } ) ) ) |
| 10 |
4 5 7 9
|
addsunif |
⊢ ( 𝜑 → ( ( 𝐴 ·s 𝐵 ) +s ( 𝐴 ·s 𝐶 ) ) = ( ( { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) } ) 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) } ∪ { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) } ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) } ) |s ( { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) } ) 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) } ∪ { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) } ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) } ) ) ) |
| 11 |
|
unab |
⊢ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ) = { 𝑎 ∣ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ∨ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) } |
| 12 |
|
rexun |
⊢ ( ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) } ) 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ↔ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) } 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ∨ ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) } 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 13 |
|
eqeq1 |
⊢ ( 𝑏 = 𝑡 → ( 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ↔ 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ) ) |
| 14 |
13
|
2rexbidv |
⊢ ( 𝑏 = 𝑡 → ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ) ) |
| 15 |
14
|
rexab |
⊢ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) } 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ↔ ∃ 𝑡 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 16 |
|
rexcom4 |
⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑡 ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑡 ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 17 |
|
rexcom4 |
⊢ ( ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑡 ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 18 |
|
ovex |
⊢ ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ∈ V |
| 19 |
|
oveq1 |
⊢ ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) → ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 20 |
19
|
eqeq2d |
⊢ ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) → ( 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ↔ 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 21 |
18 20
|
ceqsexv |
⊢ ( ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 22 |
21
|
rexbii |
⊢ ( ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 23 |
17 22
|
bitr3i |
⊢ ( ∃ 𝑡 ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 24 |
23
|
rexbii |
⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑡 ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 25 |
|
r19.41vv |
⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 26 |
25
|
exbii |
⊢ ( ∃ 𝑡 ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑡 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 27 |
16 24 26
|
3bitr3ri |
⊢ ( ∃ 𝑡 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 28 |
15 27
|
bitri |
⊢ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) } 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 29 |
|
eqeq1 |
⊢ ( 𝑏 = 𝑡 → ( 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ↔ 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ) ) |
| 30 |
29
|
2rexbidv |
⊢ ( 𝑏 = 𝑡 → ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ) ) |
| 31 |
30
|
rexab |
⊢ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) } 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ↔ ∃ 𝑡 ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 32 |
|
rexcom4 |
⊢ ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑡 ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑡 ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 33 |
|
rexcom4 |
⊢ ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑡 ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 34 |
|
ovex |
⊢ ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ∈ V |
| 35 |
|
oveq1 |
⊢ ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) → ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 36 |
35
|
eqeq2d |
⊢ ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) → ( 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ↔ 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 37 |
34 36
|
ceqsexv |
⊢ ( ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 38 |
37
|
rexbii |
⊢ ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 39 |
33 38
|
bitr3i |
⊢ ( ∃ 𝑡 ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 40 |
39
|
rexbii |
⊢ ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑡 ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 41 |
|
r19.41vv |
⊢ ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 42 |
41
|
exbii |
⊢ ( ∃ 𝑡 ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑡 ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 43 |
32 40 42
|
3bitr3ri |
⊢ ( ∃ 𝑡 ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 44 |
31 43
|
bitri |
⊢ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) } 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 45 |
28 44
|
orbi12i |
⊢ ( ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) } 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ∨ ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) } 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ∨ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 46 |
12 45
|
bitr2i |
⊢ ( ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ∨ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) } ) 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) |
| 47 |
46
|
abbii |
⊢ { 𝑎 ∣ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ∨ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) } = { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) } ) 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) } |
| 48 |
11 47
|
eqtri |
⊢ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ) = { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) } ) 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) } |
| 49 |
|
unab |
⊢ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) } ) = { 𝑎 ∣ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) ∨ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) ) } |
| 50 |
|
rexun |
⊢ ( ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) } ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ↔ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) } 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ∨ ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) } 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 51 |
|
eqeq1 |
⊢ ( 𝑏 = 𝑡 → ( 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ↔ 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) ) |
| 52 |
51
|
2rexbidv |
⊢ ( 𝑏 = 𝑡 → ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) ) |
| 53 |
52
|
rexab |
⊢ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) } 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ↔ ∃ 𝑡 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 54 |
|
rexcom4 |
⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑡 ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑡 ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 55 |
|
rexcom4 |
⊢ ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑡 ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 56 |
|
ovex |
⊢ ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ∈ V |
| 57 |
|
oveq2 |
⊢ ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) → ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) ) |
| 58 |
57
|
eqeq2d |
⊢ ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) → ( 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ↔ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) ) ) |
| 59 |
56 58
|
ceqsexv |
⊢ ( ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) ) |
| 60 |
59
|
rexbii |
⊢ ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) ) |
| 61 |
55 60
|
bitr3i |
⊢ ( ∃ 𝑡 ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) ) |
| 62 |
61
|
rexbii |
⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑡 ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) ) |
| 63 |
|
r19.41vv |
⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 64 |
63
|
exbii |
⊢ ( ∃ 𝑡 ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑡 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 65 |
54 62 64
|
3bitr3ri |
⊢ ( ∃ 𝑡 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) ) |
| 66 |
53 65
|
bitri |
⊢ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) } 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) ) |
| 67 |
|
eqeq1 |
⊢ ( 𝑏 = 𝑡 → ( 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ↔ 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) ) |
| 68 |
67
|
2rexbidv |
⊢ ( 𝑏 = 𝑡 → ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) ) |
| 69 |
68
|
rexab |
⊢ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) } 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ↔ ∃ 𝑡 ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 70 |
|
rexcom4 |
⊢ ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑡 ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑡 ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 71 |
|
rexcom4 |
⊢ ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑡 ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 72 |
|
ovex |
⊢ ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ∈ V |
| 73 |
|
oveq2 |
⊢ ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) → ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) ) |
| 74 |
73
|
eqeq2d |
⊢ ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) → ( 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ↔ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) ) ) |
| 75 |
72 74
|
ceqsexv |
⊢ ( ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) ) |
| 76 |
75
|
rexbii |
⊢ ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) ) |
| 77 |
71 76
|
bitr3i |
⊢ ( ∃ 𝑡 ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) ) |
| 78 |
77
|
rexbii |
⊢ ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑡 ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) ) |
| 79 |
|
r19.41vv |
⊢ ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 80 |
79
|
exbii |
⊢ ( ∃ 𝑡 ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑡 ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 81 |
70 78 80
|
3bitr3ri |
⊢ ( ∃ 𝑡 ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) ) |
| 82 |
69 81
|
bitri |
⊢ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) } 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) ) |
| 83 |
66 82
|
orbi12i |
⊢ ( ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) } 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ∨ ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) } 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) ∨ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) ) ) |
| 84 |
50 83
|
bitr2i |
⊢ ( ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) ∨ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) ) ↔ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) } ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) |
| 85 |
84
|
abbii |
⊢ { 𝑎 ∣ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) ∨ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) ) } = { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) } ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) } |
| 86 |
49 85
|
eqtri |
⊢ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) } ) = { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) } ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) } |
| 87 |
48 86
|
uneq12i |
⊢ ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) } ) ) = ( { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) } ) 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) } ∪ { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) } ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) } ) |
| 88 |
|
unab |
⊢ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ) = { 𝑎 ∣ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ∨ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) } |
| 89 |
|
rexun |
⊢ ( ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) } ) 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ↔ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) } 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ∨ ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) } 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 90 |
|
eqeq1 |
⊢ ( 𝑏 = 𝑡 → ( 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ↔ 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ) ) |
| 91 |
90
|
2rexbidv |
⊢ ( 𝑏 = 𝑡 → ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ) ) |
| 92 |
91
|
rexab |
⊢ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) } 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ↔ ∃ 𝑡 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 93 |
|
rexcom4 |
⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑡 ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑡 ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 94 |
|
rexcom4 |
⊢ ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑡 ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 95 |
|
ovex |
⊢ ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ∈ V |
| 96 |
|
oveq1 |
⊢ ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) → ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 97 |
96
|
eqeq2d |
⊢ ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) → ( 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ↔ 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 98 |
95 97
|
ceqsexv |
⊢ ( ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 99 |
98
|
rexbii |
⊢ ( ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 100 |
94 99
|
bitr3i |
⊢ ( ∃ 𝑡 ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 101 |
100
|
rexbii |
⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑡 ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 102 |
|
r19.41vv |
⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 103 |
102
|
exbii |
⊢ ( ∃ 𝑡 ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑡 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 104 |
93 101 103
|
3bitr3ri |
⊢ ( ∃ 𝑡 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 105 |
92 104
|
bitri |
⊢ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) } 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 106 |
|
eqeq1 |
⊢ ( 𝑏 = 𝑡 → ( 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ↔ 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ) ) |
| 107 |
106
|
2rexbidv |
⊢ ( 𝑏 = 𝑡 → ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ) ) |
| 108 |
107
|
rexab |
⊢ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) } 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ↔ ∃ 𝑡 ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 109 |
|
rexcom4 |
⊢ ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑡 ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑡 ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 110 |
|
rexcom4 |
⊢ ( ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑡 ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 111 |
|
ovex |
⊢ ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ∈ V |
| 112 |
|
oveq1 |
⊢ ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) → ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 113 |
112
|
eqeq2d |
⊢ ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) → ( 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ↔ 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 114 |
111 113
|
ceqsexv |
⊢ ( ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 115 |
114
|
rexbii |
⊢ ( ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 116 |
110 115
|
bitr3i |
⊢ ( ∃ 𝑡 ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 117 |
116
|
rexbii |
⊢ ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑡 ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 118 |
|
r19.41vv |
⊢ ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 119 |
118
|
exbii |
⊢ ( ∃ 𝑡 ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑡 ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 120 |
109 117 119
|
3bitr3ri |
⊢ ( ∃ 𝑡 ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) ∧ 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 121 |
108 120
|
bitri |
⊢ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) } 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) |
| 122 |
105 121
|
orbi12i |
⊢ ( ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) } 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ∨ ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) } 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ∨ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) ) |
| 123 |
89 122
|
bitr2i |
⊢ ( ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ∨ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) ↔ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) } ) 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) ) |
| 124 |
123
|
abbii |
⊢ { 𝑎 ∣ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) ∨ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) ) } = { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) } ) 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) } |
| 125 |
88 124
|
eqtri |
⊢ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ) = { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) } ) 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) } |
| 126 |
|
unab |
⊢ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) } ) = { 𝑎 ∣ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) ∨ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) ) } |
| 127 |
|
rexun |
⊢ ( ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) } ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ↔ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) } 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ∨ ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) } 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 128 |
|
eqeq1 |
⊢ ( 𝑏 = 𝑡 → ( 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ↔ 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) ) |
| 129 |
128
|
2rexbidv |
⊢ ( 𝑏 = 𝑡 → ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) ) |
| 130 |
129
|
rexab |
⊢ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) } 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ↔ ∃ 𝑡 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 131 |
|
rexcom4 |
⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑡 ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑡 ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 132 |
|
rexcom4 |
⊢ ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑡 ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 133 |
|
ovex |
⊢ ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ∈ V |
| 134 |
|
oveq2 |
⊢ ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) → ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) ) |
| 135 |
134
|
eqeq2d |
⊢ ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) → ( 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ↔ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) ) ) |
| 136 |
133 135
|
ceqsexv |
⊢ ( ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) ) |
| 137 |
136
|
rexbii |
⊢ ( ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) ) |
| 138 |
132 137
|
bitr3i |
⊢ ( ∃ 𝑡 ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) ) |
| 139 |
138
|
rexbii |
⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑡 ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) ) |
| 140 |
|
r19.41vv |
⊢ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 141 |
140
|
exbii |
⊢ ( ∃ 𝑡 ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑡 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 142 |
131 139 141
|
3bitr3ri |
⊢ ( ∃ 𝑡 ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) ) |
| 143 |
130 142
|
bitri |
⊢ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) } 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ↔ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) ) |
| 144 |
|
eqeq1 |
⊢ ( 𝑏 = 𝑡 → ( 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ↔ 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) ) |
| 145 |
144
|
2rexbidv |
⊢ ( 𝑏 = 𝑡 → ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) ) |
| 146 |
145
|
rexab |
⊢ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) } 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ↔ ∃ 𝑡 ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 147 |
|
rexcom4 |
⊢ ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑡 ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑡 ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 148 |
|
rexcom4 |
⊢ ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑡 ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 149 |
|
ovex |
⊢ ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ∈ V |
| 150 |
|
oveq2 |
⊢ ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) → ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) ) |
| 151 |
150
|
eqeq2d |
⊢ ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) → ( 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ↔ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) ) ) |
| 152 |
149 151
|
ceqsexv |
⊢ ( ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) ) |
| 153 |
152
|
rexbii |
⊢ ( ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) ∃ 𝑡 ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) ) |
| 154 |
148 153
|
bitr3i |
⊢ ( ∃ 𝑡 ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) ) |
| 155 |
154
|
rexbii |
⊢ ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑡 ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) ) |
| 156 |
|
r19.41vv |
⊢ ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 157 |
156
|
exbii |
⊢ ( ∃ 𝑡 ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) ( 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑡 ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ) |
| 158 |
147 155 157
|
3bitr3ri |
⊢ ( ∃ 𝑡 ( ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑡 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ∧ 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) ) |
| 159 |
146 158
|
bitri |
⊢ ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) } 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ↔ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) ) |
| 160 |
143 159
|
orbi12i |
⊢ ( ( ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) } 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ∨ ∃ 𝑡 ∈ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) } 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) ↔ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) ∨ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) ) ) |
| 161 |
127 160
|
bitr2i |
⊢ ( ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) ∨ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) ) ↔ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) } ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) ) |
| 162 |
161
|
abbii |
⊢ { 𝑎 ∣ ( ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) ∨ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) ) } = { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) } ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) } |
| 163 |
126 162
|
eqtri |
⊢ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) } ) = { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) } ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) } |
| 164 |
125 163
|
uneq12i |
⊢ ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) } ) ) = ( { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) } ) 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) } ∪ { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) } ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) } ) |
| 165 |
87 164
|
oveq12i |
⊢ ( ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) } ) ) |s ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) } ) ) ) = ( ( { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) } ) 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) } ∪ { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) } ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) } ) |s ( { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) } ) 𝑎 = ( 𝑡 +s ( 𝐴 ·s 𝐶 ) ) } ∪ { 𝑎 ∣ ∃ 𝑡 ∈ ( { 𝑏 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) } ∪ { 𝑏 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑏 = ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) } ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s 𝑡 ) } ) ) |
| 166 |
10 165
|
eqtr4di |
⊢ ( 𝜑 → ( ( 𝐴 ·s 𝐵 ) +s ( 𝐴 ·s 𝐶 ) ) = ( ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝐿 ·s 𝑧𝐿 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝑅 ·s 𝑧𝑅 ) ) ) } ) ) |s ( ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑦𝑅 ∈ ( R ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝐿 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑦𝑅 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑦𝐿 ∈ ( L ‘ 𝐵 ) 𝑎 = ( ( ( ( 𝑥𝑅 ·s 𝐵 ) +s ( 𝐴 ·s 𝑦𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑦𝐿 ) ) +s ( 𝐴 ·s 𝐶 ) ) } ) ∪ ( { 𝑎 ∣ ∃ 𝑥𝐿 ∈ ( L ‘ 𝐴 ) ∃ 𝑧𝑅 ∈ ( R ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝐿 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝑅 ) ) -s ( 𝑥𝐿 ·s 𝑧𝑅 ) ) ) } ∪ { 𝑎 ∣ ∃ 𝑥𝑅 ∈ ( R ‘ 𝐴 ) ∃ 𝑧𝐿 ∈ ( L ‘ 𝐶 ) 𝑎 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝑥𝑅 ·s 𝐶 ) +s ( 𝐴 ·s 𝑧𝐿 ) ) -s ( 𝑥𝑅 ·s 𝑧𝐿 ) ) ) } ) ) ) ) |