| Step |
Hyp |
Ref |
Expression |
| 1 |
|
mulscl |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( 𝐴 ·s 𝐵 ) ∈ No ) |
| 2 |
1
|
adantr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( ∃ 𝑛 ∈ ℕs ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ) ∧ ( ∃ 𝑚 ∈ ℕs ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) ) ) → ( 𝐴 ·s 𝐵 ) ∈ No ) |
| 3 |
|
remulscllem2 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( 𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs ) ∧ ( ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ) ) ) → ∃ 𝑝 ∈ ℕs ( ( -us ‘ 𝑝 ) <s ( 𝐴 ·s 𝐵 ) ∧ ( 𝐴 ·s 𝐵 ) <s 𝑝 ) ) |
| 4 |
3
|
expr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝑛 ∈ ℕs ∧ 𝑚 ∈ ℕs ) ) → ( ( ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ) → ∃ 𝑝 ∈ ℕs ( ( -us ‘ 𝑝 ) <s ( 𝐴 ·s 𝐵 ) ∧ ( 𝐴 ·s 𝐵 ) <s 𝑝 ) ) ) |
| 5 |
4
|
rexlimdvva |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ∃ 𝑛 ∈ ℕs ∃ 𝑚 ∈ ℕs ( ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ) → ∃ 𝑝 ∈ ℕs ( ( -us ‘ 𝑝 ) <s ( 𝐴 ·s 𝐵 ) ∧ ( 𝐴 ·s 𝐵 ) <s 𝑝 ) ) ) |
| 6 |
|
simpl |
⊢ ( ( ∃ 𝑛 ∈ ℕs ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ) → ∃ 𝑛 ∈ ℕs ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ) |
| 7 |
|
simpl |
⊢ ( ( ∃ 𝑚 ∈ ℕs ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) → ∃ 𝑚 ∈ ℕs ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ) |
| 8 |
6 7
|
anim12i |
⊢ ( ( ( ∃ 𝑛 ∈ ℕs ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ) ∧ ( ∃ 𝑚 ∈ ℕs ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) ) → ( ∃ 𝑛 ∈ ℕs ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ ∃ 𝑚 ∈ ℕs ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ) ) |
| 9 |
|
reeanv |
⊢ ( ∃ 𝑛 ∈ ℕs ∃ 𝑚 ∈ ℕs ( ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ) ↔ ( ∃ 𝑛 ∈ ℕs ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ ∃ 𝑚 ∈ ℕs ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ) ) |
| 10 |
8 9
|
sylibr |
⊢ ( ( ( ∃ 𝑛 ∈ ℕs ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ) ∧ ( ∃ 𝑚 ∈ ℕs ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) ) → ∃ 𝑛 ∈ ℕs ∃ 𝑚 ∈ ℕs ( ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ) ) |
| 11 |
5 10
|
impel |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( ∃ 𝑛 ∈ ℕs ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ) ∧ ( ∃ 𝑚 ∈ ℕs ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) ) ) → ∃ 𝑝 ∈ ℕs ( ( -us ‘ 𝑝 ) <s ( 𝐴 ·s 𝐵 ) ∧ ( 𝐴 ·s 𝐵 ) <s 𝑝 ) ) |
| 12 |
|
simpr |
⊢ ( ( ∃ 𝑛 ∈ ℕs ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ) → 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ) |
| 13 |
|
simpr |
⊢ ( ( ∃ 𝑚 ∈ ℕs ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) → 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) |
| 14 |
12 13
|
anim12i |
⊢ ( ( ( ∃ 𝑛 ∈ ℕs ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ) ∧ ( ∃ 𝑚 ∈ ℕs ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) ) → ( 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) ) |
| 15 |
|
recut |
⊢ ( 𝐴 ∈ No → { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } <<s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) |
| 16 |
15
|
adantr |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } <<s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) |
| 17 |
16
|
adantr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) ) → { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } <<s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) |
| 18 |
|
recut |
⊢ ( 𝐵 ∈ No → { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } <<s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) |
| 19 |
18
|
ad2antlr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) ) → { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } <<s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) |
| 20 |
|
simprl |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) ) → 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ) |
| 21 |
|
simprr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) ) → 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) |
| 22 |
17 19 20 21
|
mulsunif2 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) ) → ( 𝐴 ·s 𝐵 ) = ( ( { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) } ∪ { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) } ) |s ( { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) } ∪ { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) } ) ) ) |
| 23 |
|
r19.41v |
⊢ ( ∃ 𝑛 ∈ ℕs ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ( ∃ 𝑛 ∈ ℕs 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 24 |
23
|
exbii |
⊢ ( ∃ 𝑡 ∃ 𝑛 ∈ ℕs ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑡 ( ∃ 𝑛 ∈ ℕs 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 25 |
|
rexcom4 |
⊢ ( ∃ 𝑛 ∈ ℕs ∃ 𝑡 ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑡 ∃ 𝑛 ∈ ℕs ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 26 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑡 → ( 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ↔ 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ) |
| 27 |
26
|
rexbidv |
⊢ ( 𝑥 = 𝑡 → ( ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ↔ ∃ 𝑛 ∈ ℕs 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ) |
| 28 |
27
|
rexab |
⊢ ( ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) ↔ ∃ 𝑡 ( ∃ 𝑛 ∈ ℕs 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 29 |
24 25 28
|
3bitr4ri |
⊢ ( ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) ↔ ∃ 𝑛 ∈ ℕs ∃ 𝑡 ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 30 |
|
ovex |
⊢ ( 𝐴 -s ( 1s /su 𝑛 ) ) ∈ V |
| 31 |
|
oveq2 |
⊢ ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) → ( 𝐴 -s 𝑡 ) = ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ) |
| 32 |
31
|
oveq1d |
⊢ ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) → ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) = ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) |
| 33 |
32
|
oveq2d |
⊢ ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) → ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ) |
| 34 |
33
|
eqeq2d |
⊢ ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) → ( 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 35 |
34
|
rexbidv |
⊢ ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) → ( ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) ↔ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 36 |
30 35
|
ceqsexv |
⊢ ( ∃ 𝑡 ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ) |
| 37 |
|
r19.41v |
⊢ ( ∃ 𝑚 ∈ ℕs ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ( ∃ 𝑚 ∈ ℕs 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 38 |
37
|
exbii |
⊢ ( ∃ 𝑢 ∃ 𝑚 ∈ ℕs ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑢 ( ∃ 𝑚 ∈ ℕs 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 39 |
|
rexcom4 |
⊢ ( ∃ 𝑚 ∈ ℕs ∃ 𝑢 ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑢 ∃ 𝑚 ∈ ℕs ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 40 |
|
eqeq1 |
⊢ ( 𝑦 = 𝑢 → ( 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ↔ 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ) ) |
| 41 |
40
|
rexbidv |
⊢ ( 𝑦 = 𝑢 → ( ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ↔ ∃ 𝑚 ∈ ℕs 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ) ) |
| 42 |
41
|
rexab |
⊢ ( ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ↔ ∃ 𝑢 ( ∃ 𝑚 ∈ ℕs 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 43 |
38 39 42
|
3bitr4ri |
⊢ ( ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ↔ ∃ 𝑚 ∈ ℕs ∃ 𝑢 ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 44 |
36 43
|
bitri |
⊢ ( ∃ 𝑡 ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑚 ∈ ℕs ∃ 𝑢 ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 45 |
|
ovex |
⊢ ( 𝐵 -s ( 1s /su 𝑚 ) ) ∈ V |
| 46 |
|
oveq2 |
⊢ ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) → ( 𝐵 -s 𝑢 ) = ( 𝐵 -s ( 𝐵 -s ( 1s /su 𝑚 ) ) ) ) |
| 47 |
46
|
oveq2d |
⊢ ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) → ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) = ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s ( 𝐵 -s ( 1s /su 𝑚 ) ) ) ) ) |
| 48 |
47
|
oveq2d |
⊢ ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) → ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s ( 𝐵 -s ( 1s /su 𝑚 ) ) ) ) ) ) |
| 49 |
48
|
eqeq2d |
⊢ ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) → ( 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s ( 𝐵 -s ( 1s /su 𝑚 ) ) ) ) ) ) ) |
| 50 |
45 49
|
ceqsexv |
⊢ ( ∃ 𝑢 ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s ( 𝐵 -s ( 1s /su 𝑚 ) ) ) ) ) ) |
| 51 |
|
simplll |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → 𝐴 ∈ No ) |
| 52 |
|
1sno |
⊢ 1s ∈ No |
| 53 |
52
|
a1i |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → 1s ∈ No ) |
| 54 |
|
nnsno |
⊢ ( 𝑛 ∈ ℕs → 𝑛 ∈ No ) |
| 55 |
54
|
ad2antlr |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → 𝑛 ∈ No ) |
| 56 |
|
nnne0s |
⊢ ( 𝑛 ∈ ℕs → 𝑛 ≠ 0s ) |
| 57 |
56
|
ad2antlr |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → 𝑛 ≠ 0s ) |
| 58 |
53 55 57
|
divscld |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( 1s /su 𝑛 ) ∈ No ) |
| 59 |
51 58
|
nncansd |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) = ( 1s /su 𝑛 ) ) |
| 60 |
|
simpllr |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → 𝐵 ∈ No ) |
| 61 |
|
nnsno |
⊢ ( 𝑚 ∈ ℕs → 𝑚 ∈ No ) |
| 62 |
61
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → 𝑚 ∈ No ) |
| 63 |
|
nnne0s |
⊢ ( 𝑚 ∈ ℕs → 𝑚 ≠ 0s ) |
| 64 |
63
|
adantl |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → 𝑚 ≠ 0s ) |
| 65 |
53 62 64
|
divscld |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( 1s /su 𝑚 ) ∈ No ) |
| 66 |
60 65
|
nncansd |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( 𝐵 -s ( 𝐵 -s ( 1s /su 𝑚 ) ) ) = ( 1s /su 𝑚 ) ) |
| 67 |
59 66
|
oveq12d |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s ( 𝐵 -s ( 1s /su 𝑚 ) ) ) ) = ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) |
| 68 |
67
|
oveq2d |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s ( 𝐵 -s ( 1s /su 𝑚 ) ) ) ) ) = ( ( 𝐴 ·s 𝐵 ) -s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) |
| 69 |
68
|
eqeq2d |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s ( 𝐵 -s ( 1s /su 𝑚 ) ) ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 70 |
50 69
|
bitrid |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( ∃ 𝑢 ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 71 |
70
|
rexbidva |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) → ( ∃ 𝑚 ∈ ℕs ∃ 𝑢 ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑚 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 72 |
44 71
|
bitrid |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) → ( ∃ 𝑡 ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑚 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 73 |
72
|
rexbidva |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ∃ 𝑛 ∈ ℕs ∃ 𝑡 ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑛 ∈ ℕs ∃ 𝑚 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 74 |
|
remulscllem1 |
⊢ ( ∃ 𝑛 ∈ ℕs ∃ 𝑚 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ↔ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) ) |
| 75 |
73 74
|
bitrdi |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ∃ 𝑛 ∈ ℕs ∃ 𝑡 ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) ) ) |
| 76 |
29 75
|
bitrid |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) ↔ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) ) ) |
| 77 |
76
|
abbidv |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) } = { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) } ) |
| 78 |
|
r19.41v |
⊢ ( ∃ 𝑛 ∈ ℕs ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ( ∃ 𝑛 ∈ ℕs 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 79 |
78
|
exbii |
⊢ ( ∃ 𝑡 ∃ 𝑛 ∈ ℕs ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑡 ( ∃ 𝑛 ∈ ℕs 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 80 |
|
rexcom4 |
⊢ ( ∃ 𝑛 ∈ ℕs ∃ 𝑡 ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑡 ∃ 𝑛 ∈ ℕs ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 81 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑡 → ( 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ↔ 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ) ) |
| 82 |
81
|
rexbidv |
⊢ ( 𝑥 = 𝑡 → ( ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ↔ ∃ 𝑛 ∈ ℕs 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ) ) |
| 83 |
82
|
rexab |
⊢ ( ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ↔ ∃ 𝑡 ( ∃ 𝑛 ∈ ℕs 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 84 |
79 80 83
|
3bitr4ri |
⊢ ( ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ↔ ∃ 𝑛 ∈ ℕs ∃ 𝑡 ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 85 |
|
ovex |
⊢ ( 𝐴 +s ( 1s /su 𝑛 ) ) ∈ V |
| 86 |
|
oveq1 |
⊢ ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) → ( 𝑡 -s 𝐴 ) = ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ) |
| 87 |
86
|
oveq1d |
⊢ ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) → ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) = ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) |
| 88 |
87
|
oveq2d |
⊢ ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) → ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) |
| 89 |
88
|
eqeq2d |
⊢ ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) → ( 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 90 |
89
|
rexbidv |
⊢ ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) → ( ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ↔ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 91 |
85 90
|
ceqsexv |
⊢ ( ∃ 𝑡 ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) |
| 92 |
|
r19.41v |
⊢ ( ∃ 𝑚 ∈ ℕs ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ( ∃ 𝑚 ∈ ℕs 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 93 |
92
|
exbii |
⊢ ( ∃ 𝑢 ∃ 𝑚 ∈ ℕs ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑢 ( ∃ 𝑚 ∈ ℕs 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 94 |
|
rexcom4 |
⊢ ( ∃ 𝑚 ∈ ℕs ∃ 𝑢 ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑢 ∃ 𝑚 ∈ ℕs ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 95 |
|
eqeq1 |
⊢ ( 𝑦 = 𝑢 → ( 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ↔ 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ) ) |
| 96 |
95
|
rexbidv |
⊢ ( 𝑦 = 𝑢 → ( ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ↔ ∃ 𝑚 ∈ ℕs 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ) ) |
| 97 |
96
|
rexab |
⊢ ( ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ↔ ∃ 𝑢 ( ∃ 𝑚 ∈ ℕs 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 98 |
93 94 97
|
3bitr4ri |
⊢ ( ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ↔ ∃ 𝑚 ∈ ℕs ∃ 𝑢 ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 99 |
91 98
|
bitri |
⊢ ( ∃ 𝑡 ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑚 ∈ ℕs ∃ 𝑢 ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 100 |
|
ovex |
⊢ ( 𝐵 +s ( 1s /su 𝑚 ) ) ∈ V |
| 101 |
|
oveq1 |
⊢ ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) → ( 𝑢 -s 𝐵 ) = ( ( 𝐵 +s ( 1s /su 𝑚 ) ) -s 𝐵 ) ) |
| 102 |
101
|
oveq2d |
⊢ ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) → ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) = ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( ( 𝐵 +s ( 1s /su 𝑚 ) ) -s 𝐵 ) ) ) |
| 103 |
102
|
oveq2d |
⊢ ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) → ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( ( 𝐵 +s ( 1s /su 𝑚 ) ) -s 𝐵 ) ) ) ) |
| 104 |
103
|
eqeq2d |
⊢ ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) → ( 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( ( 𝐵 +s ( 1s /su 𝑚 ) ) -s 𝐵 ) ) ) ) ) |
| 105 |
100 104
|
ceqsexv |
⊢ ( ∃ 𝑢 ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( ( 𝐵 +s ( 1s /su 𝑚 ) ) -s 𝐵 ) ) ) ) |
| 106 |
|
pncan2s |
⊢ ( ( 𝐴 ∈ No ∧ ( 1s /su 𝑛 ) ∈ No ) → ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) = ( 1s /su 𝑛 ) ) |
| 107 |
51 58 106
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) = ( 1s /su 𝑛 ) ) |
| 108 |
|
pncan2s |
⊢ ( ( 𝐵 ∈ No ∧ ( 1s /su 𝑚 ) ∈ No ) → ( ( 𝐵 +s ( 1s /su 𝑚 ) ) -s 𝐵 ) = ( 1s /su 𝑚 ) ) |
| 109 |
60 65 108
|
syl2anc |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( ( 𝐵 +s ( 1s /su 𝑚 ) ) -s 𝐵 ) = ( 1s /su 𝑚 ) ) |
| 110 |
107 109
|
oveq12d |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( ( 𝐵 +s ( 1s /su 𝑚 ) ) -s 𝐵 ) ) = ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) |
| 111 |
110
|
oveq2d |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( ( 𝐵 +s ( 1s /su 𝑚 ) ) -s 𝐵 ) ) ) = ( ( 𝐴 ·s 𝐵 ) -s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) |
| 112 |
111
|
eqeq2d |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( ( 𝐵 +s ( 1s /su 𝑚 ) ) -s 𝐵 ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 113 |
105 112
|
bitrid |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( ∃ 𝑢 ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 114 |
113
|
rexbidva |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) → ( ∃ 𝑚 ∈ ℕs ∃ 𝑢 ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑚 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 115 |
99 114
|
bitrid |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) → ( ∃ 𝑡 ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑚 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 116 |
115
|
rexbidva |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ∃ 𝑛 ∈ ℕs ∃ 𝑡 ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑛 ∈ ℕs ∃ 𝑚 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 117 |
116 74
|
bitrdi |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ∃ 𝑛 ∈ ℕs ∃ 𝑡 ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) ) ) |
| 118 |
84 117
|
bitrid |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) ↔ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) ) ) |
| 119 |
118
|
abbidv |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) } = { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) } ) |
| 120 |
77 119
|
uneq12d |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) } ∪ { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) } ) = ( { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) } ∪ { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) } ) ) |
| 121 |
|
unidm |
⊢ ( { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) } ∪ { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) } ) = { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) } |
| 122 |
120 121
|
eqtrdi |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) } ∪ { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) } ) = { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) } ) |
| 123 |
|
r19.41v |
⊢ ( ∃ 𝑛 ∈ ℕs ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ( ∃ 𝑛 ∈ ℕs 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 124 |
123
|
exbii |
⊢ ( ∃ 𝑡 ∃ 𝑛 ∈ ℕs ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑡 ( ∃ 𝑛 ∈ ℕs 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 125 |
|
rexcom4 |
⊢ ( ∃ 𝑛 ∈ ℕs ∃ 𝑡 ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑡 ∃ 𝑛 ∈ ℕs ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 126 |
27
|
rexab |
⊢ ( ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) ↔ ∃ 𝑡 ( ∃ 𝑛 ∈ ℕs 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 127 |
124 125 126
|
3bitr4ri |
⊢ ( ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) ↔ ∃ 𝑛 ∈ ℕs ∃ 𝑡 ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 128 |
31
|
oveq1d |
⊢ ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) → ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) = ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) |
| 129 |
128
|
oveq2d |
⊢ ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) → ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ) |
| 130 |
129
|
eqeq2d |
⊢ ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) → ( 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 131 |
130
|
rexbidv |
⊢ ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) → ( ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) ↔ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 132 |
30 131
|
ceqsexv |
⊢ ( ∃ 𝑡 ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ) |
| 133 |
|
r19.41v |
⊢ ( ∃ 𝑚 ∈ ℕs ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ( ∃ 𝑚 ∈ ℕs 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 134 |
133
|
exbii |
⊢ ( ∃ 𝑢 ∃ 𝑚 ∈ ℕs ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑢 ( ∃ 𝑚 ∈ ℕs 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 135 |
|
rexcom4 |
⊢ ( ∃ 𝑚 ∈ ℕs ∃ 𝑢 ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑢 ∃ 𝑚 ∈ ℕs ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 136 |
96
|
rexab |
⊢ ( ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ↔ ∃ 𝑢 ( ∃ 𝑚 ∈ ℕs 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 137 |
134 135 136
|
3bitr4ri |
⊢ ( ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ↔ ∃ 𝑚 ∈ ℕs ∃ 𝑢 ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 138 |
132 137
|
bitri |
⊢ ( ∃ 𝑡 ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑚 ∈ ℕs ∃ 𝑢 ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ) |
| 139 |
101
|
oveq2d |
⊢ ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) → ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) = ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( ( 𝐵 +s ( 1s /su 𝑚 ) ) -s 𝐵 ) ) ) |
| 140 |
139
|
oveq2d |
⊢ ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) → ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( ( 𝐵 +s ( 1s /su 𝑚 ) ) -s 𝐵 ) ) ) ) |
| 141 |
140
|
eqeq2d |
⊢ ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) → ( 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( ( 𝐵 +s ( 1s /su 𝑚 ) ) -s 𝐵 ) ) ) ) ) |
| 142 |
100 141
|
ceqsexv |
⊢ ( ∃ 𝑢 ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( ( 𝐵 +s ( 1s /su 𝑚 ) ) -s 𝐵 ) ) ) ) |
| 143 |
59 109
|
oveq12d |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( ( 𝐵 +s ( 1s /su 𝑚 ) ) -s 𝐵 ) ) = ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) |
| 144 |
143
|
oveq2d |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( ( 𝐵 +s ( 1s /su 𝑚 ) ) -s 𝐵 ) ) ) = ( ( 𝐴 ·s 𝐵 ) +s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) |
| 145 |
144
|
eqeq2d |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( ( 𝐵 +s ( 1s /su 𝑚 ) ) -s 𝐵 ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 146 |
142 145
|
bitrid |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( ∃ 𝑢 ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 147 |
146
|
rexbidva |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) → ( ∃ 𝑚 ∈ ℕs ∃ 𝑢 ( 𝑢 = ( 𝐵 +s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s ( 𝐴 -s ( 1s /su 𝑛 ) ) ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑚 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 148 |
138 147
|
bitrid |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) → ( ∃ 𝑡 ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑚 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 149 |
148
|
rexbidva |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ∃ 𝑛 ∈ ℕs ∃ 𝑡 ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑛 ∈ ℕs ∃ 𝑚 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 150 |
|
remulscllem1 |
⊢ ( ∃ 𝑛 ∈ ℕs ∃ 𝑚 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ↔ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) ) |
| 151 |
149 150
|
bitrdi |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ∃ 𝑛 ∈ ℕs ∃ 𝑡 ( 𝑡 = ( 𝐴 -s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) ) ↔ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) ) ) |
| 152 |
127 151
|
bitrid |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) ↔ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) ) ) |
| 153 |
152
|
abbidv |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) } = { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) } ) |
| 154 |
|
r19.41v |
⊢ ( ∃ 𝑛 ∈ ℕs ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ( ∃ 𝑛 ∈ ℕs 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 155 |
154
|
exbii |
⊢ ( ∃ 𝑡 ∃ 𝑛 ∈ ℕs ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑡 ( ∃ 𝑛 ∈ ℕs 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 156 |
|
rexcom4 |
⊢ ( ∃ 𝑛 ∈ ℕs ∃ 𝑡 ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑡 ∃ 𝑛 ∈ ℕs ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 157 |
82
|
rexab |
⊢ ( ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ↔ ∃ 𝑡 ( ∃ 𝑛 ∈ ℕs 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 158 |
155 156 157
|
3bitr4ri |
⊢ ( ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ↔ ∃ 𝑛 ∈ ℕs ∃ 𝑡 ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 159 |
86
|
oveq1d |
⊢ ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) → ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) = ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) |
| 160 |
159
|
oveq2d |
⊢ ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) → ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) |
| 161 |
160
|
eqeq2d |
⊢ ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) → ( 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 162 |
161
|
rexbidv |
⊢ ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) → ( ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ↔ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 163 |
85 162
|
ceqsexv |
⊢ ( ∃ 𝑡 ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) |
| 164 |
|
r19.41v |
⊢ ( ∃ 𝑚 ∈ ℕs ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ( ∃ 𝑚 ∈ ℕs 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 165 |
164
|
exbii |
⊢ ( ∃ 𝑢 ∃ 𝑚 ∈ ℕs ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑢 ( ∃ 𝑚 ∈ ℕs 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 166 |
|
rexcom4 |
⊢ ( ∃ 𝑚 ∈ ℕs ∃ 𝑢 ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑢 ∃ 𝑚 ∈ ℕs ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 167 |
41
|
rexab |
⊢ ( ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ↔ ∃ 𝑢 ( ∃ 𝑚 ∈ ℕs 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 168 |
165 166 167
|
3bitr4ri |
⊢ ( ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ↔ ∃ 𝑚 ∈ ℕs ∃ 𝑢 ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 169 |
163 168
|
bitri |
⊢ ( ∃ 𝑡 ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑚 ∈ ℕs ∃ 𝑢 ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ) |
| 170 |
46
|
oveq2d |
⊢ ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) → ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) = ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s ( 𝐵 -s ( 1s /su 𝑚 ) ) ) ) ) |
| 171 |
170
|
oveq2d |
⊢ ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) → ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s ( 𝐵 -s ( 1s /su 𝑚 ) ) ) ) ) ) |
| 172 |
171
|
eqeq2d |
⊢ ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) → ( 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s ( 𝐵 -s ( 1s /su 𝑚 ) ) ) ) ) ) ) |
| 173 |
45 172
|
ceqsexv |
⊢ ( ∃ 𝑢 ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s ( 𝐵 -s ( 1s /su 𝑚 ) ) ) ) ) ) |
| 174 |
107 66
|
oveq12d |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s ( 𝐵 -s ( 1s /su 𝑚 ) ) ) ) = ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) |
| 175 |
174
|
oveq2d |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s ( 𝐵 -s ( 1s /su 𝑚 ) ) ) ) ) = ( ( 𝐴 ·s 𝐵 ) +s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) |
| 176 |
175
|
eqeq2d |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s ( 𝐵 -s ( 1s /su 𝑚 ) ) ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 177 |
173 176
|
bitrid |
⊢ ( ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) ∧ 𝑚 ∈ ℕs ) → ( ∃ 𝑢 ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 178 |
177
|
rexbidva |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) → ( ∃ 𝑚 ∈ ℕs ∃ 𝑢 ( 𝑢 = ( 𝐵 -s ( 1s /su 𝑚 ) ) ∧ 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( ( 𝐴 +s ( 1s /su 𝑛 ) ) -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑚 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 179 |
169 178
|
bitrid |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ 𝑛 ∈ ℕs ) → ( ∃ 𝑡 ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑚 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 180 |
179
|
rexbidva |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ∃ 𝑛 ∈ ℕs ∃ 𝑡 ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑛 ∈ ℕs ∃ 𝑚 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 1s /su 𝑛 ) ·s ( 1s /su 𝑚 ) ) ) ) ) |
| 181 |
180 150
|
bitrdi |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ∃ 𝑛 ∈ ℕs ∃ 𝑡 ( 𝑡 = ( 𝐴 +s ( 1s /su 𝑛 ) ) ∧ ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ) ↔ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) ) ) |
| 182 |
158 181
|
bitrid |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) ↔ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) ) ) |
| 183 |
182
|
abbidv |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) } = { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) } ) |
| 184 |
153 183
|
uneq12d |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) } ∪ { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) } ) = ( { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) } ∪ { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) } ) ) |
| 185 |
|
unidm |
⊢ ( { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) } ∪ { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) } ) = { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) } |
| 186 |
184 185
|
eqtrdi |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) } ∪ { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) } ) = { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) } ) |
| 187 |
122 186
|
oveq12d |
⊢ ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( ( { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) } ∪ { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) } ) |s ( { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) } ∪ { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) } ) ) = ( { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) } |s { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) } ) ) |
| 188 |
187
|
adantr |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) ) → ( ( { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝐴 -s 𝑡 ) ·s ( 𝐵 -s 𝑢 ) ) ) } ∪ { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( ( 𝑡 -s 𝐴 ) ·s ( 𝑢 -s 𝐵 ) ) ) } ) |s ( { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝐴 -s 𝑡 ) ·s ( 𝑢 -s 𝐵 ) ) ) } ∪ { 𝑧 ∣ ∃ 𝑡 ∈ { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ∃ 𝑢 ∈ { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( ( 𝑡 -s 𝐴 ) ·s ( 𝐵 -s 𝑢 ) ) ) } ) ) = ( { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) } |s { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) } ) ) |
| 189 |
22 188
|
eqtrd |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) ) → ( 𝐴 ·s 𝐵 ) = ( { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) } |s { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) } ) ) |
| 190 |
14 189
|
sylan2 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( ∃ 𝑛 ∈ ℕs ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ) ∧ ( ∃ 𝑚 ∈ ℕs ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) ) ) → ( 𝐴 ·s 𝐵 ) = ( { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) } |s { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) } ) ) |
| 191 |
2 11 190
|
jca32 |
⊢ ( ( ( 𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ ( ( ∃ 𝑛 ∈ ℕs ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ) ∧ ( ∃ 𝑚 ∈ ℕs ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) ) ) → ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ∃ 𝑝 ∈ ℕs ( ( -us ‘ 𝑝 ) <s ( 𝐴 ·s 𝐵 ) ∧ ( 𝐴 ·s 𝐵 ) <s 𝑝 ) ∧ ( 𝐴 ·s 𝐵 ) = ( { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) } |s { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) } ) ) ) ) |
| 192 |
191
|
an4s |
⊢ ( ( ( 𝐴 ∈ No ∧ ( ∃ 𝑛 ∈ ℕs ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ) ) ∧ ( 𝐵 ∈ No ∧ ( ∃ 𝑚 ∈ ℕs ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) ) ) → ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ∃ 𝑝 ∈ ℕs ( ( -us ‘ 𝑝 ) <s ( 𝐴 ·s 𝐵 ) ∧ ( 𝐴 ·s 𝐵 ) <s 𝑝 ) ∧ ( 𝐴 ·s 𝐵 ) = ( { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) } |s { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) } ) ) ) ) |
| 193 |
|
elreno |
⊢ ( 𝐴 ∈ ℝs ↔ ( 𝐴 ∈ No ∧ ( ∃ 𝑛 ∈ ℕs ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ) ) ) |
| 194 |
|
elreno |
⊢ ( 𝐵 ∈ ℝs ↔ ( 𝐵 ∈ No ∧ ( ∃ 𝑚 ∈ ℕs ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) ) ) |
| 195 |
193 194
|
anbi12i |
⊢ ( ( 𝐴 ∈ ℝs ∧ 𝐵 ∈ ℝs ) ↔ ( ( 𝐴 ∈ No ∧ ( ∃ 𝑛 ∈ ℕs ( ( -us ‘ 𝑛 ) <s 𝐴 ∧ 𝐴 <s 𝑛 ) ∧ 𝐴 = ( { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 -s ( 1s /su 𝑛 ) ) } |s { 𝑥 ∣ ∃ 𝑛 ∈ ℕs 𝑥 = ( 𝐴 +s ( 1s /su 𝑛 ) ) } ) ) ) ∧ ( 𝐵 ∈ No ∧ ( ∃ 𝑚 ∈ ℕs ( ( -us ‘ 𝑚 ) <s 𝐵 ∧ 𝐵 <s 𝑚 ) ∧ 𝐵 = ( { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 -s ( 1s /su 𝑚 ) ) } |s { 𝑦 ∣ ∃ 𝑚 ∈ ℕs 𝑦 = ( 𝐵 +s ( 1s /su 𝑚 ) ) } ) ) ) ) ) |
| 196 |
|
elreno |
⊢ ( ( 𝐴 ·s 𝐵 ) ∈ ℝs ↔ ( ( 𝐴 ·s 𝐵 ) ∈ No ∧ ( ∃ 𝑝 ∈ ℕs ( ( -us ‘ 𝑝 ) <s ( 𝐴 ·s 𝐵 ) ∧ ( 𝐴 ·s 𝐵 ) <s 𝑝 ) ∧ ( 𝐴 ·s 𝐵 ) = ( { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) -s ( 1s /su 𝑝 ) ) } |s { 𝑧 ∣ ∃ 𝑝 ∈ ℕs 𝑧 = ( ( 𝐴 ·s 𝐵 ) +s ( 1s /su 𝑝 ) ) } ) ) ) ) |
| 197 |
192 195 196
|
3imtr4i |
⊢ ( ( 𝐴 ∈ ℝs ∧ 𝐵 ∈ ℝs ) → ( 𝐴 ·s 𝐵 ) ∈ ℝs ) |