| Step |
Hyp |
Ref |
Expression |
| 1 |
|
mulsproplem.1 |
⊢ ( 𝜑 → ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 2 |
|
mulsproplem1.1 |
⊢ ( 𝜑 → 𝑋 ∈ No ) |
| 3 |
|
mulsproplem1.2 |
⊢ ( 𝜑 → 𝑌 ∈ No ) |
| 4 |
|
mulsproplem1.3 |
⊢ ( 𝜑 → 𝑍 ∈ No ) |
| 5 |
|
mulsproplem1.4 |
⊢ ( 𝜑 → 𝑊 ∈ No ) |
| 6 |
|
mulsproplem1.5 |
⊢ ( 𝜑 → 𝑇 ∈ No ) |
| 7 |
|
mulsproplem1.6 |
⊢ ( 𝜑 → 𝑈 ∈ No ) |
| 8 |
|
mulsproplem1.7 |
⊢ ( 𝜑 → ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑈 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑈 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ) |
| 9 |
|
fveq2 |
⊢ ( 𝑎 = 𝑋 → ( bday ‘ 𝑎 ) = ( bday ‘ 𝑋 ) ) |
| 10 |
9
|
oveq1d |
⊢ ( 𝑎 = 𝑋 → ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) = ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑏 ) ) ) |
| 11 |
10
|
uneq1d |
⊢ ( 𝑎 = 𝑋 → ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) = ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ) |
| 12 |
11
|
eleq1d |
⊢ ( 𝑎 = 𝑋 → ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ↔ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ) ) |
| 13 |
|
oveq1 |
⊢ ( 𝑎 = 𝑋 → ( 𝑎 ·s 𝑏 ) = ( 𝑋 ·s 𝑏 ) ) |
| 14 |
13
|
eleq1d |
⊢ ( 𝑎 = 𝑋 → ( ( 𝑎 ·s 𝑏 ) ∈ No ↔ ( 𝑋 ·s 𝑏 ) ∈ No ) ) |
| 15 |
14
|
anbi1d |
⊢ ( 𝑎 = 𝑋 → ( ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ↔ ( ( 𝑋 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 16 |
12 15
|
imbi12d |
⊢ ( 𝑎 = 𝑋 → ( ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ( ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝑋 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) ) |
| 17 |
|
fveq2 |
⊢ ( 𝑏 = 𝑌 → ( bday ‘ 𝑏 ) = ( bday ‘ 𝑌 ) ) |
| 18 |
17
|
oveq2d |
⊢ ( 𝑏 = 𝑌 → ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑏 ) ) = ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ) |
| 19 |
18
|
uneq1d |
⊢ ( 𝑏 = 𝑌 → ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) = ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ) |
| 20 |
19
|
eleq1d |
⊢ ( 𝑏 = 𝑌 → ( ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ↔ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ) ) |
| 21 |
|
oveq2 |
⊢ ( 𝑏 = 𝑌 → ( 𝑋 ·s 𝑏 ) = ( 𝑋 ·s 𝑌 ) ) |
| 22 |
21
|
eleq1d |
⊢ ( 𝑏 = 𝑌 → ( ( 𝑋 ·s 𝑏 ) ∈ No ↔ ( 𝑋 ·s 𝑌 ) ∈ No ) ) |
| 23 |
22
|
anbi1d |
⊢ ( 𝑏 = 𝑌 → ( ( ( 𝑋 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ↔ ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 24 |
20 23
|
imbi12d |
⊢ ( 𝑏 = 𝑌 → ( ( ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝑋 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ( ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) ) |
| 25 |
|
fveq2 |
⊢ ( 𝑐 = 𝑍 → ( bday ‘ 𝑐 ) = ( bday ‘ 𝑍 ) ) |
| 26 |
25
|
oveq1d |
⊢ ( 𝑐 = 𝑍 → ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) = ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ) |
| 27 |
26
|
uneq1d |
⊢ ( 𝑐 = 𝑍 → ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) = ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ) |
| 28 |
25
|
oveq1d |
⊢ ( 𝑐 = 𝑍 → ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) = ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ) |
| 29 |
28
|
uneq1d |
⊢ ( 𝑐 = 𝑍 → ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) = ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) |
| 30 |
27 29
|
uneq12d |
⊢ ( 𝑐 = 𝑍 → ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) = ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) |
| 31 |
30
|
uneq2d |
⊢ ( 𝑐 = 𝑍 → ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) = ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ) |
| 32 |
31
|
eleq1d |
⊢ ( 𝑐 = 𝑍 → ( ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ↔ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ) ) |
| 33 |
|
breq1 |
⊢ ( 𝑐 = 𝑍 → ( 𝑐 <s 𝑑 ↔ 𝑍 <s 𝑑 ) ) |
| 34 |
33
|
anbi1d |
⊢ ( 𝑐 = 𝑍 → ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) ↔ ( 𝑍 <s 𝑑 ∧ 𝑒 <s 𝑓 ) ) ) |
| 35 |
|
oveq1 |
⊢ ( 𝑐 = 𝑍 → ( 𝑐 ·s 𝑓 ) = ( 𝑍 ·s 𝑓 ) ) |
| 36 |
|
oveq1 |
⊢ ( 𝑐 = 𝑍 → ( 𝑐 ·s 𝑒 ) = ( 𝑍 ·s 𝑒 ) ) |
| 37 |
35 36
|
oveq12d |
⊢ ( 𝑐 = 𝑍 → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) = ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑒 ) ) ) |
| 38 |
37
|
breq1d |
⊢ ( 𝑐 = 𝑍 → ( ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ↔ ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) |
| 39 |
34 38
|
imbi12d |
⊢ ( 𝑐 = 𝑍 → ( ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ↔ ( ( 𝑍 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) |
| 40 |
39
|
anbi2d |
⊢ ( 𝑐 = 𝑍 → ( ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ↔ ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑍 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 41 |
32 40
|
imbi12d |
⊢ ( 𝑐 = 𝑍 → ( ( ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ( ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑍 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) ) |
| 42 |
|
fveq2 |
⊢ ( 𝑑 = 𝑊 → ( bday ‘ 𝑑 ) = ( bday ‘ 𝑊 ) ) |
| 43 |
42
|
oveq1d |
⊢ ( 𝑑 = 𝑊 → ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) = ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) |
| 44 |
43
|
uneq2d |
⊢ ( 𝑑 = 𝑊 → ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) = ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) ) |
| 45 |
42
|
oveq1d |
⊢ ( 𝑑 = 𝑊 → ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) = ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑒 ) ) ) |
| 46 |
45
|
uneq2d |
⊢ ( 𝑑 = 𝑊 → ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) = ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑒 ) ) ) ) |
| 47 |
44 46
|
uneq12d |
⊢ ( 𝑑 = 𝑊 → ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) = ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑒 ) ) ) ) ) |
| 48 |
47
|
uneq2d |
⊢ ( 𝑑 = 𝑊 → ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) = ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ) |
| 49 |
48
|
eleq1d |
⊢ ( 𝑑 = 𝑊 → ( ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ↔ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ) ) |
| 50 |
|
breq2 |
⊢ ( 𝑑 = 𝑊 → ( 𝑍 <s 𝑑 ↔ 𝑍 <s 𝑊 ) ) |
| 51 |
50
|
anbi1d |
⊢ ( 𝑑 = 𝑊 → ( ( 𝑍 <s 𝑑 ∧ 𝑒 <s 𝑓 ) ↔ ( 𝑍 <s 𝑊 ∧ 𝑒 <s 𝑓 ) ) ) |
| 52 |
|
oveq1 |
⊢ ( 𝑑 = 𝑊 → ( 𝑑 ·s 𝑓 ) = ( 𝑊 ·s 𝑓 ) ) |
| 53 |
|
oveq1 |
⊢ ( 𝑑 = 𝑊 → ( 𝑑 ·s 𝑒 ) = ( 𝑊 ·s 𝑒 ) ) |
| 54 |
52 53
|
oveq12d |
⊢ ( 𝑑 = 𝑊 → ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) = ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑒 ) ) ) |
| 55 |
54
|
breq2d |
⊢ ( 𝑑 = 𝑊 → ( ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ↔ ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑒 ) ) <s ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑒 ) ) ) ) |
| 56 |
51 55
|
imbi12d |
⊢ ( 𝑑 = 𝑊 → ( ( ( 𝑍 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ↔ ( ( 𝑍 <s 𝑊 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑒 ) ) <s ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑒 ) ) ) ) ) |
| 57 |
56
|
anbi2d |
⊢ ( 𝑑 = 𝑊 → ( ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑍 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ↔ ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑍 <s 𝑊 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑒 ) ) <s ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑒 ) ) ) ) ) ) |
| 58 |
49 57
|
imbi12d |
⊢ ( 𝑑 = 𝑊 → ( ( ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑍 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ( ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑍 <s 𝑊 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑒 ) ) <s ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑒 ) ) ) ) ) ) ) |
| 59 |
|
fveq2 |
⊢ ( 𝑒 = 𝑇 → ( bday ‘ 𝑒 ) = ( bday ‘ 𝑇 ) ) |
| 60 |
59
|
oveq2d |
⊢ ( 𝑒 = 𝑇 → ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) = ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ) |
| 61 |
60
|
uneq1d |
⊢ ( 𝑒 = 𝑇 → ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) = ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) ) |
| 62 |
59
|
oveq2d |
⊢ ( 𝑒 = 𝑇 → ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑒 ) ) = ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) |
| 63 |
62
|
uneq2d |
⊢ ( 𝑒 = 𝑇 → ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑒 ) ) ) = ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) ) |
| 64 |
61 63
|
uneq12d |
⊢ ( 𝑒 = 𝑇 → ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑒 ) ) ) ) = ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) ) ) |
| 65 |
64
|
uneq2d |
⊢ ( 𝑒 = 𝑇 → ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑒 ) ) ) ) ) = ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) ) ) ) |
| 66 |
65
|
eleq1d |
⊢ ( 𝑒 = 𝑇 → ( ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ↔ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ) ) |
| 67 |
|
breq1 |
⊢ ( 𝑒 = 𝑇 → ( 𝑒 <s 𝑓 ↔ 𝑇 <s 𝑓 ) ) |
| 68 |
67
|
anbi2d |
⊢ ( 𝑒 = 𝑇 → ( ( 𝑍 <s 𝑊 ∧ 𝑒 <s 𝑓 ) ↔ ( 𝑍 <s 𝑊 ∧ 𝑇 <s 𝑓 ) ) ) |
| 69 |
|
oveq2 |
⊢ ( 𝑒 = 𝑇 → ( 𝑍 ·s 𝑒 ) = ( 𝑍 ·s 𝑇 ) ) |
| 70 |
69
|
oveq2d |
⊢ ( 𝑒 = 𝑇 → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑒 ) ) = ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑇 ) ) ) |
| 71 |
|
oveq2 |
⊢ ( 𝑒 = 𝑇 → ( 𝑊 ·s 𝑒 ) = ( 𝑊 ·s 𝑇 ) ) |
| 72 |
71
|
oveq2d |
⊢ ( 𝑒 = 𝑇 → ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑒 ) ) = ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑇 ) ) ) |
| 73 |
70 72
|
breq12d |
⊢ ( 𝑒 = 𝑇 → ( ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑒 ) ) <s ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑒 ) ) ↔ ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑇 ) ) <s ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑇 ) ) ) ) |
| 74 |
68 73
|
imbi12d |
⊢ ( 𝑒 = 𝑇 → ( ( ( 𝑍 <s 𝑊 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑒 ) ) <s ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑒 ) ) ) ↔ ( ( 𝑍 <s 𝑊 ∧ 𝑇 <s 𝑓 ) → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑇 ) ) <s ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑇 ) ) ) ) ) |
| 75 |
74
|
anbi2d |
⊢ ( 𝑒 = 𝑇 → ( ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑍 <s 𝑊 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑒 ) ) <s ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑒 ) ) ) ) ↔ ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑍 <s 𝑊 ∧ 𝑇 <s 𝑓 ) → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑇 ) ) <s ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑇 ) ) ) ) ) ) |
| 76 |
66 75
|
imbi12d |
⊢ ( 𝑒 = 𝑇 → ( ( ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑍 <s 𝑊 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑒 ) ) <s ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑒 ) ) ) ) ) ↔ ( ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑍 <s 𝑊 ∧ 𝑇 <s 𝑓 ) → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑇 ) ) <s ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑇 ) ) ) ) ) ) ) |
| 77 |
|
fveq2 |
⊢ ( 𝑓 = 𝑈 → ( bday ‘ 𝑓 ) = ( bday ‘ 𝑈 ) ) |
| 78 |
77
|
oveq2d |
⊢ ( 𝑓 = 𝑈 → ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) = ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑈 ) ) ) |
| 79 |
78
|
uneq2d |
⊢ ( 𝑓 = 𝑈 → ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) = ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑈 ) ) ) ) |
| 80 |
77
|
oveq2d |
⊢ ( 𝑓 = 𝑈 → ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) = ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑈 ) ) ) |
| 81 |
80
|
uneq1d |
⊢ ( 𝑓 = 𝑈 → ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) = ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑈 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) ) |
| 82 |
79 81
|
uneq12d |
⊢ ( 𝑓 = 𝑈 → ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) ) = ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑈 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑈 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) ) ) |
| 83 |
82
|
uneq2d |
⊢ ( 𝑓 = 𝑈 → ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) ) ) = ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑈 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑈 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) ) ) ) |
| 84 |
83
|
eleq1d |
⊢ ( 𝑓 = 𝑈 → ( ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ↔ ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑈 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑈 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) ) ) |
| 85 |
|
breq2 |
⊢ ( 𝑓 = 𝑈 → ( 𝑇 <s 𝑓 ↔ 𝑇 <s 𝑈 ) ) |
| 86 |
85
|
anbi2d |
⊢ ( 𝑓 = 𝑈 → ( ( 𝑍 <s 𝑊 ∧ 𝑇 <s 𝑓 ) ↔ ( 𝑍 <s 𝑊 ∧ 𝑇 <s 𝑈 ) ) ) |
| 87 |
|
oveq2 |
⊢ ( 𝑓 = 𝑈 → ( 𝑍 ·s 𝑓 ) = ( 𝑍 ·s 𝑈 ) ) |
| 88 |
87
|
oveq1d |
⊢ ( 𝑓 = 𝑈 → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑇 ) ) = ( ( 𝑍 ·s 𝑈 ) -s ( 𝑍 ·s 𝑇 ) ) ) |
| 89 |
|
oveq2 |
⊢ ( 𝑓 = 𝑈 → ( 𝑊 ·s 𝑓 ) = ( 𝑊 ·s 𝑈 ) ) |
| 90 |
89
|
oveq1d |
⊢ ( 𝑓 = 𝑈 → ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑇 ) ) = ( ( 𝑊 ·s 𝑈 ) -s ( 𝑊 ·s 𝑇 ) ) ) |
| 91 |
88 90
|
breq12d |
⊢ ( 𝑓 = 𝑈 → ( ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑇 ) ) <s ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑇 ) ) ↔ ( ( 𝑍 ·s 𝑈 ) -s ( 𝑍 ·s 𝑇 ) ) <s ( ( 𝑊 ·s 𝑈 ) -s ( 𝑊 ·s 𝑇 ) ) ) ) |
| 92 |
86 91
|
imbi12d |
⊢ ( 𝑓 = 𝑈 → ( ( ( 𝑍 <s 𝑊 ∧ 𝑇 <s 𝑓 ) → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑇 ) ) <s ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑇 ) ) ) ↔ ( ( 𝑍 <s 𝑊 ∧ 𝑇 <s 𝑈 ) → ( ( 𝑍 ·s 𝑈 ) -s ( 𝑍 ·s 𝑇 ) ) <s ( ( 𝑊 ·s 𝑈 ) -s ( 𝑊 ·s 𝑇 ) ) ) ) ) |
| 93 |
92
|
anbi2d |
⊢ ( 𝑓 = 𝑈 → ( ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑍 <s 𝑊 ∧ 𝑇 <s 𝑓 ) → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑇 ) ) <s ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑇 ) ) ) ) ↔ ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑍 <s 𝑊 ∧ 𝑇 <s 𝑈 ) → ( ( 𝑍 ·s 𝑈 ) -s ( 𝑍 ·s 𝑇 ) ) <s ( ( 𝑊 ·s 𝑈 ) -s ( 𝑊 ·s 𝑇 ) ) ) ) ) ) |
| 94 |
84 93
|
imbi12d |
⊢ ( 𝑓 = 𝑈 → ( ( ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑍 <s 𝑊 ∧ 𝑇 <s 𝑓 ) → ( ( 𝑍 ·s 𝑓 ) -s ( 𝑍 ·s 𝑇 ) ) <s ( ( 𝑊 ·s 𝑓 ) -s ( 𝑊 ·s 𝑇 ) ) ) ) ) ↔ ( ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑈 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑈 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑍 <s 𝑊 ∧ 𝑇 <s 𝑈 ) → ( ( 𝑍 ·s 𝑈 ) -s ( 𝑍 ·s 𝑇 ) ) <s ( ( 𝑊 ·s 𝑈 ) -s ( 𝑊 ·s 𝑇 ) ) ) ) ) ) ) |
| 95 |
16 24 41 58 76 94
|
rspc6v |
⊢ ( ( ( 𝑋 ∈ No ∧ 𝑌 ∈ No ) ∧ ( 𝑍 ∈ No ∧ 𝑊 ∈ No ) ∧ ( 𝑇 ∈ No ∧ 𝑈 ∈ No ) ) → ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) → ( ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑈 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑈 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑍 <s 𝑊 ∧ 𝑇 <s 𝑈 ) → ( ( 𝑍 ·s 𝑈 ) -s ( 𝑍 ·s 𝑇 ) ) <s ( ( 𝑊 ·s 𝑈 ) -s ( 𝑊 ·s 𝑇 ) ) ) ) ) ) ) |
| 96 |
2 3 4 5 6 7 95
|
syl222anc |
⊢ ( 𝜑 → ( ∀ 𝑎 ∈ No ∀ 𝑏 ∈ No ∀ 𝑐 ∈ No ∀ 𝑑 ∈ No ∀ 𝑒 ∈ No ∀ 𝑓 ∈ No ( ( ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝑎 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) → ( ( ( ( bday ‘ 𝑋 ) +no ( bday ‘ 𝑌 ) ) ∪ ( ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑇 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑈 ) ) ) ∪ ( ( ( bday ‘ 𝑍 ) +no ( bday ‘ 𝑈 ) ) ∪ ( ( bday ‘ 𝑊 ) +no ( bday ‘ 𝑇 ) ) ) ) ) ∈ ( ( ( bday ‘ 𝐴 ) +no ( bday ‘ 𝐵 ) ) ∪ ( ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐸 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐹 ) ) ) ∪ ( ( ( bday ‘ 𝐶 ) +no ( bday ‘ 𝐹 ) ) ∪ ( ( bday ‘ 𝐷 ) +no ( bday ‘ 𝐸 ) ) ) ) ) → ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑍 <s 𝑊 ∧ 𝑇 <s 𝑈 ) → ( ( 𝑍 ·s 𝑈 ) -s ( 𝑍 ·s 𝑇 ) ) <s ( ( 𝑊 ·s 𝑈 ) -s ( 𝑊 ·s 𝑇 ) ) ) ) ) ) ) |
| 97 |
1 8 96
|
mp2d |
⊢ ( 𝜑 → ( ( 𝑋 ·s 𝑌 ) ∈ No ∧ ( ( 𝑍 <s 𝑊 ∧ 𝑇 <s 𝑈 ) → ( ( 𝑍 ·s 𝑈 ) -s ( 𝑍 ·s 𝑇 ) ) <s ( ( 𝑊 ·s 𝑈 ) -s ( 𝑊 ·s 𝑇 ) ) ) ) ) |