| 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 |
|
fveq2 |
⊢ ( 𝑎 = 𝑔 → ( bday ‘ 𝑎 ) = ( bday ‘ 𝑔 ) ) |
| 3 |
2
|
oveq1d |
⊢ ( 𝑎 = 𝑔 → ( ( bday ‘ 𝑎 ) +no ( bday ‘ 𝑏 ) ) = ( ( bday ‘ 𝑔 ) +no ( bday ‘ 𝑏 ) ) ) |
| 4 |
3
|
uneq1d |
⊢ ( 𝑎 = 𝑔 → ( ( ( 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 ‘ 𝑔 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ) |
| 5 |
4
|
eleq1d |
⊢ ( 𝑎 = 𝑔 → ( ( ( ( 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 ‘ 𝑔 ) +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 ) ) ) ) ) ) ) |
| 6 |
|
oveq1 |
⊢ ( 𝑎 = 𝑔 → ( 𝑎 ·s 𝑏 ) = ( 𝑔 ·s 𝑏 ) ) |
| 7 |
6
|
eleq1d |
⊢ ( 𝑎 = 𝑔 → ( ( 𝑎 ·s 𝑏 ) ∈ No ↔ ( 𝑔 ·s 𝑏 ) ∈ No ) ) |
| 8 |
7
|
anbi1d |
⊢ ( 𝑎 = 𝑔 → ( ( ( 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 ) ) ) ) ↔ ( ( 𝑔 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 9 |
5 8
|
imbi12d |
⊢ ( 𝑎 = 𝑔 → ( ( ( ( ( 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 ‘ 𝑔 ) +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 ) ) ) ) ) → ( ( 𝑔 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) ) |
| 10 |
|
fveq2 |
⊢ ( 𝑏 = ℎ → ( bday ‘ 𝑏 ) = ( bday ‘ ℎ ) ) |
| 11 |
10
|
oveq2d |
⊢ ( 𝑏 = ℎ → ( ( bday ‘ 𝑔 ) +no ( bday ‘ 𝑏 ) ) = ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ) |
| 12 |
11
|
uneq1d |
⊢ ( 𝑏 = ℎ → ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ 𝑏 ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ) |
| 13 |
12
|
eleq1d |
⊢ ( 𝑏 = ℎ → ( ( ( ( bday ‘ 𝑔 ) +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 ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( 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 ) ) ) ) ) ) ) |
| 14 |
|
oveq2 |
⊢ ( 𝑏 = ℎ → ( 𝑔 ·s 𝑏 ) = ( 𝑔 ·s ℎ ) ) |
| 15 |
14
|
eleq1d |
⊢ ( 𝑏 = ℎ → ( ( 𝑔 ·s 𝑏 ) ∈ No ↔ ( 𝑔 ·s ℎ ) ∈ No ) ) |
| 16 |
15
|
anbi1d |
⊢ ( 𝑏 = ℎ → ( ( ( 𝑔 ·s 𝑏 ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ↔ ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 17 |
13 16
|
imbi12d |
⊢ ( 𝑏 = ℎ → ( ( ( ( ( bday ‘ 𝑔 ) +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 ) ) ) ) ) → ( ( 𝑔 ·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 ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( 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 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) ) |
| 18 |
|
fveq2 |
⊢ ( 𝑐 = 𝑖 → ( bday ‘ 𝑐 ) = ( bday ‘ 𝑖 ) ) |
| 19 |
18
|
oveq1d |
⊢ ( 𝑐 = 𝑖 → ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) = ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ) |
| 20 |
19
|
uneq1d |
⊢ ( 𝑐 = 𝑖 → ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) = ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ) |
| 21 |
18
|
oveq1d |
⊢ ( 𝑐 = 𝑖 → ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) = ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ) |
| 22 |
21
|
uneq1d |
⊢ ( 𝑐 = 𝑖 → ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) = ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) |
| 23 |
20 22
|
uneq12d |
⊢ ( 𝑐 = 𝑖 → ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) = ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) |
| 24 |
23
|
uneq2d |
⊢ ( 𝑐 = 𝑖 → ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑐 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ) |
| 25 |
24
|
eleq1d |
⊢ ( 𝑐 = 𝑖 → ( ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( 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 ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +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 ) ) ) ) ) ) ) |
| 26 |
|
breq1 |
⊢ ( 𝑐 = 𝑖 → ( 𝑐 <s 𝑑 ↔ 𝑖 <s 𝑑 ) ) |
| 27 |
26
|
anbi1d |
⊢ ( 𝑐 = 𝑖 → ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) ↔ ( 𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓 ) ) ) |
| 28 |
|
oveq1 |
⊢ ( 𝑐 = 𝑖 → ( 𝑐 ·s 𝑓 ) = ( 𝑖 ·s 𝑓 ) ) |
| 29 |
|
oveq1 |
⊢ ( 𝑐 = 𝑖 → ( 𝑐 ·s 𝑒 ) = ( 𝑖 ·s 𝑒 ) ) |
| 30 |
28 29
|
oveq12d |
⊢ ( 𝑐 = 𝑖 → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) = ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) ) |
| 31 |
30
|
breq1d |
⊢ ( 𝑐 = 𝑖 → ( ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ↔ ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) |
| 32 |
27 31
|
imbi12d |
⊢ ( 𝑐 = 𝑖 → ( ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ↔ ( ( 𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) |
| 33 |
32
|
anbi2d |
⊢ ( 𝑐 = 𝑖 → ( ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ↔ ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) |
| 34 |
25 33
|
imbi12d |
⊢ ( 𝑐 = 𝑖 → ( ( ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( 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 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑐 ·s 𝑓 ) -s ( 𝑐 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ( ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +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 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ) ) |
| 35 |
|
fveq2 |
⊢ ( 𝑑 = 𝑗 → ( bday ‘ 𝑑 ) = ( bday ‘ 𝑗 ) ) |
| 36 |
35
|
oveq1d |
⊢ ( 𝑑 = 𝑗 → ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) = ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) |
| 37 |
36
|
uneq2d |
⊢ ( 𝑑 = 𝑗 → ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) = ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ) |
| 38 |
35
|
oveq1d |
⊢ ( 𝑑 = 𝑗 → ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) = ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) ) |
| 39 |
38
|
uneq2d |
⊢ ( 𝑑 = 𝑗 → ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) = ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) ) ) |
| 40 |
37 39
|
uneq12d |
⊢ ( 𝑑 = 𝑗 → ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) = ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) ) ) ) |
| 41 |
40
|
uneq2d |
⊢ ( 𝑑 = 𝑗 → ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑒 ) ) ) ) ) = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) ) ) ) ) |
| 42 |
41
|
eleq1d |
⊢ ( 𝑑 = 𝑗 → ( ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +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 ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +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 ) ) ) ) ) ) ) |
| 43 |
|
breq2 |
⊢ ( 𝑑 = 𝑗 → ( 𝑖 <s 𝑑 ↔ 𝑖 <s 𝑗 ) ) |
| 44 |
43
|
anbi1d |
⊢ ( 𝑑 = 𝑗 → ( ( 𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓 ) ↔ ( 𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓 ) ) ) |
| 45 |
|
oveq1 |
⊢ ( 𝑑 = 𝑗 → ( 𝑑 ·s 𝑓 ) = ( 𝑗 ·s 𝑓 ) ) |
| 46 |
|
oveq1 |
⊢ ( 𝑑 = 𝑗 → ( 𝑑 ·s 𝑒 ) = ( 𝑗 ·s 𝑒 ) ) |
| 47 |
45 46
|
oveq12d |
⊢ ( 𝑑 = 𝑗 → ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) = ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) ) |
| 48 |
47
|
breq2d |
⊢ ( 𝑑 = 𝑗 → ( ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ↔ ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) ) ) |
| 49 |
44 48
|
imbi12d |
⊢ ( 𝑑 = 𝑗 → ( ( ( 𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ↔ ( ( 𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) ) ) ) |
| 50 |
49
|
anbi2d |
⊢ ( 𝑑 = 𝑗 → ( ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ↔ ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) ) ) ) ) |
| 51 |
42 50
|
imbi12d |
⊢ ( 𝑑 = 𝑗 → ( ( ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑑 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +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 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑑 ·s 𝑓 ) -s ( 𝑑 ·s 𝑒 ) ) ) ) ) ↔ ( ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +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 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) ) ) ) ) ) |
| 52 |
|
fveq2 |
⊢ ( 𝑒 = 𝑘 → ( bday ‘ 𝑒 ) = ( bday ‘ 𝑘 ) ) |
| 53 |
52
|
oveq2d |
⊢ ( 𝑒 = 𝑘 → ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) = ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ) |
| 54 |
53
|
uneq1d |
⊢ ( 𝑒 = 𝑘 → ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) = ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ) |
| 55 |
52
|
oveq2d |
⊢ ( 𝑒 = 𝑘 → ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) = ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) |
| 56 |
55
|
uneq2d |
⊢ ( 𝑒 = 𝑘 → ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) ) = ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) |
| 57 |
54 56
|
uneq12d |
⊢ ( 𝑒 = 𝑘 → ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) ) ) = ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) |
| 58 |
57
|
uneq2d |
⊢ ( 𝑒 = 𝑘 → ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑒 ) ) ) ) ) = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ) |
| 59 |
58
|
eleq1d |
⊢ ( 𝑒 = 𝑘 → ( ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +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 ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∈ ( ( ( 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 ) ) ) ) ) ) ) |
| 60 |
|
breq1 |
⊢ ( 𝑒 = 𝑘 → ( 𝑒 <s 𝑓 ↔ 𝑘 <s 𝑓 ) ) |
| 61 |
60
|
anbi2d |
⊢ ( 𝑒 = 𝑘 → ( ( 𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓 ) ↔ ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓 ) ) ) |
| 62 |
|
oveq2 |
⊢ ( 𝑒 = 𝑘 → ( 𝑖 ·s 𝑒 ) = ( 𝑖 ·s 𝑘 ) ) |
| 63 |
62
|
oveq2d |
⊢ ( 𝑒 = 𝑘 → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) = ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) ) |
| 64 |
|
oveq2 |
⊢ ( 𝑒 = 𝑘 → ( 𝑗 ·s 𝑒 ) = ( 𝑗 ·s 𝑘 ) ) |
| 65 |
64
|
oveq2d |
⊢ ( 𝑒 = 𝑘 → ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) = ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) ) |
| 66 |
63 65
|
breq12d |
⊢ ( 𝑒 = 𝑘 → ( ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) ↔ ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) |
| 67 |
61 66
|
imbi12d |
⊢ ( 𝑒 = 𝑘 → ( ( ( 𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) ) ↔ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) |
| 68 |
67
|
anbi2d |
⊢ ( 𝑒 = 𝑘 → ( ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) ) ) ↔ ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) |
| 69 |
59 68
|
imbi12d |
⊢ ( 𝑒 = 𝑘 → ( ( ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑒 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +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 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑒 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑒 ) ) ) ) ) ↔ ( ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∈ ( ( ( 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 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) ) |
| 70 |
|
fveq2 |
⊢ ( 𝑓 = 𝑙 → ( bday ‘ 𝑓 ) = ( bday ‘ 𝑙 ) ) |
| 71 |
70
|
oveq2d |
⊢ ( 𝑓 = 𝑙 → ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) = ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) |
| 72 |
71
|
uneq2d |
⊢ ( 𝑓 = 𝑙 → ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) = ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ) |
| 73 |
70
|
oveq2d |
⊢ ( 𝑓 = 𝑙 → ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) = ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ) |
| 74 |
73
|
uneq1d |
⊢ ( 𝑓 = 𝑙 → ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) = ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) |
| 75 |
72 74
|
uneq12d |
⊢ ( 𝑓 = 𝑙 → ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) = ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) |
| 76 |
75
|
uneq2d |
⊢ ( 𝑓 = 𝑙 → ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) = ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ) |
| 77 |
76
|
eleq1d |
⊢ ( 𝑓 = 𝑙 → ( ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∈ ( ( ( 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 ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∈ ( ( ( 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 ) ) ) ) ) ) ) |
| 78 |
|
breq2 |
⊢ ( 𝑓 = 𝑙 → ( 𝑘 <s 𝑓 ↔ 𝑘 <s 𝑙 ) ) |
| 79 |
78
|
anbi2d |
⊢ ( 𝑓 = 𝑙 → ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓 ) ↔ ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) ) ) |
| 80 |
|
oveq2 |
⊢ ( 𝑓 = 𝑙 → ( 𝑖 ·s 𝑓 ) = ( 𝑖 ·s 𝑙 ) ) |
| 81 |
80
|
oveq1d |
⊢ ( 𝑓 = 𝑙 → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) = ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) ) |
| 82 |
|
oveq2 |
⊢ ( 𝑓 = 𝑙 → ( 𝑗 ·s 𝑓 ) = ( 𝑗 ·s 𝑙 ) ) |
| 83 |
82
|
oveq1d |
⊢ ( 𝑓 = 𝑙 → ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) = ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) |
| 84 |
81 83
|
breq12d |
⊢ ( 𝑓 = 𝑙 → ( ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) ↔ ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) |
| 85 |
79 84
|
imbi12d |
⊢ ( 𝑓 = 𝑙 → ( ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) ) ↔ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) |
| 86 |
85
|
anbi2d |
⊢ ( 𝑓 = 𝑙 → ( ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ↔ ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) |
| 87 |
77 86
|
imbi12d |
⊢ ( 𝑓 = 𝑙 → ( ( ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑓 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑓 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∈ ( ( ( 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 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓 ) → ( ( 𝑖 ·s 𝑓 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑓 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ↔ ( ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∈ ( ( ( 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 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) ) |
| 88 |
9 17 34 51 69 87
|
cbvral6vw |
⊢ ( ∀ 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 ) ) ) ) ) ↔ ∀ 𝑔 ∈ No ∀ ℎ ∈ No ∀ 𝑖 ∈ No ∀ 𝑗 ∈ No ∀ 𝑘 ∈ No ∀ 𝑙 ∈ No ( ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∈ ( ( ( 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 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) |
| 89 |
1 88
|
sylib |
⊢ ( 𝜑 → ∀ 𝑔 ∈ No ∀ ℎ ∈ No ∀ 𝑖 ∈ No ∀ 𝑗 ∈ No ∀ 𝑘 ∈ No ∀ 𝑙 ∈ No ( ( ( ( bday ‘ 𝑔 ) +no ( bday ‘ ℎ ) ) ∪ ( ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑘 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑙 ) ) ) ∪ ( ( ( bday ‘ 𝑖 ) +no ( bday ‘ 𝑙 ) ) ∪ ( ( bday ‘ 𝑗 ) +no ( bday ‘ 𝑘 ) ) ) ) ) ∈ ( ( ( 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 ) ) ) ) ) → ( ( 𝑔 ·s ℎ ) ∈ No ∧ ( ( 𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙 ) → ( ( 𝑖 ·s 𝑙 ) -s ( 𝑖 ·s 𝑘 ) ) <s ( ( 𝑗 ·s 𝑙 ) -s ( 𝑗 ·s 𝑘 ) ) ) ) ) ) |