Proof of Theorem imasaddvallem
Step | Hyp | Ref
| Expression |
1 | | df-ov 6552 |
. 2
⊢ ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = ( ∙ ‘〈(𝐹‘𝑋), (𝐹‘𝑌)〉) |
2 | | imasaddf.f |
. . . . . 6
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) |
3 | | imasaddf.e |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) |
4 | | imasaddflem.a |
. . . . . 6
⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
5 | 2, 3, 4 | imasaddfnlem 16011 |
. . . . 5
⊢ (𝜑 → ∙ Fn (𝐵 × 𝐵)) |
6 | | fnfun 5902 |
. . . . 5
⊢ ( ∙ Fn
(𝐵 × 𝐵) → Fun ∙ ) |
7 | 5, 6 | syl 17 |
. . . 4
⊢ (𝜑 → Fun ∙ ) |
8 | 7 | 3ad2ant1 1075 |
. . 3
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → Fun ∙ ) |
9 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑝 = 𝑋 → (𝐹‘𝑝) = (𝐹‘𝑋)) |
10 | 9 | opeq1d 4346 |
. . . . . . . . . 10
⊢ (𝑝 = 𝑋 → 〈(𝐹‘𝑝), (𝐹‘𝑌)〉 = 〈(𝐹‘𝑋), (𝐹‘𝑌)〉) |
11 | | oveq1 6556 |
. . . . . . . . . . 11
⊢ (𝑝 = 𝑋 → (𝑝 · 𝑌) = (𝑋 · 𝑌)) |
12 | 11 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (𝑝 = 𝑋 → (𝐹‘(𝑝 · 𝑌)) = (𝐹‘(𝑋 · 𝑌))) |
13 | 10, 12 | opeq12d 4348 |
. . . . . . . . 9
⊢ (𝑝 = 𝑋 → 〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉 = 〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉) |
14 | 13 | sneqd 4137 |
. . . . . . . 8
⊢ (𝑝 = 𝑋 → {〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉} = {〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉}) |
15 | 14 | ssiun2s 4500 |
. . . . . . 7
⊢ (𝑋 ∈ 𝑉 → {〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉} ⊆ ∪ 𝑝 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉}) |
16 | 15 | 3ad2ant2 1076 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉} ⊆ ∪ 𝑝 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉}) |
17 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑞 = 𝑌 → (𝐹‘𝑞) = (𝐹‘𝑌)) |
18 | 17 | opeq2d 4347 |
. . . . . . . . . . . 12
⊢ (𝑞 = 𝑌 → 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 = 〈(𝐹‘𝑝), (𝐹‘𝑌)〉) |
19 | | oveq2 6557 |
. . . . . . . . . . . . 13
⊢ (𝑞 = 𝑌 → (𝑝 · 𝑞) = (𝑝 · 𝑌)) |
20 | 19 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (𝑞 = 𝑌 → (𝐹‘(𝑝 · 𝑞)) = (𝐹‘(𝑝 · 𝑌))) |
21 | 18, 20 | opeq12d 4348 |
. . . . . . . . . . 11
⊢ (𝑞 = 𝑌 → 〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉 = 〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉) |
22 | 21 | sneqd 4137 |
. . . . . . . . . 10
⊢ (𝑞 = 𝑌 → {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} = {〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉}) |
23 | 22 | ssiun2s 4500 |
. . . . . . . . 9
⊢ (𝑌 ∈ 𝑉 → {〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉} ⊆ ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
24 | 23 | ralrimivw 2950 |
. . . . . . . 8
⊢ (𝑌 ∈ 𝑉 → ∀𝑝 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉} ⊆ ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
25 | | ss2iun 4472 |
. . . . . . . 8
⊢
(∀𝑝 ∈
𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉} ⊆ ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} → ∪ 𝑝 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉} ⊆ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
26 | 24, 25 | syl 17 |
. . . . . . 7
⊢ (𝑌 ∈ 𝑉 → ∪
𝑝 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉} ⊆ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
27 | 26 | 3ad2ant3 1077 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∪
𝑝 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉} ⊆ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
28 | 16, 27 | sstrd 3578 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉} ⊆ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
29 | 4 | 3ad2ant1 1075 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
30 | 28, 29 | sseqtr4d 3605 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉} ⊆ ∙ ) |
31 | | opex 4859 |
. . . . 5
⊢
〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉 ∈ V |
32 | 31 | snss 4259 |
. . . 4
⊢
(〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉 ∈ ∙ ↔
{〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉} ⊆ ∙ ) |
33 | 30, 32 | sylibr 223 |
. . 3
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉 ∈ ∙ ) |
34 | | funopfv 6145 |
. . 3
⊢ (Fun
∙
→ (〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉 ∈ ∙ → ( ∙
‘〈(𝐹‘𝑋), (𝐹‘𝑌)〉) = (𝐹‘(𝑋 · 𝑌)))) |
35 | 8, 33, 34 | sylc 63 |
. 2
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ( ∙ ‘〈(𝐹‘𝑋), (𝐹‘𝑌)〉) = (𝐹‘(𝑋 · 𝑌))) |
36 | 1, 35 | syl5eq 2656 |
1
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌))) |