Proof of Theorem imasaddvallem
| Step | Hyp | Ref
| Expression |
| 1 | | df-ov 7434 |
. 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 17573 |
. . . . 5
⊢ (𝜑 → ∙ Fn (𝐵 × 𝐵)) |
| 6 | | fnfun 6668 |
. . . . 5
⊢ ( ∙ Fn
(𝐵 × 𝐵) → Fun ∙ ) |
| 7 | 5, 6 | syl 17 |
. . . 4
⊢ (𝜑 → Fun ∙ ) |
| 8 | 7 | 3ad2ant1 1134 |
. . 3
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → Fun ∙ ) |
| 9 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑝 = 𝑋 → (𝐹‘𝑝) = (𝐹‘𝑋)) |
| 10 | 9 | opeq1d 4879 |
. . . . . . . . . 10
⊢ (𝑝 = 𝑋 → 〈(𝐹‘𝑝), (𝐹‘𝑌)〉 = 〈(𝐹‘𝑋), (𝐹‘𝑌)〉) |
| 11 | | fvoveq1 7454 |
. . . . . . . . . 10
⊢ (𝑝 = 𝑋 → (𝐹‘(𝑝 · 𝑌)) = (𝐹‘(𝑋 · 𝑌))) |
| 12 | 10, 11 | opeq12d 4881 |
. . . . . . . . 9
⊢ (𝑝 = 𝑋 → 〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉 = 〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉) |
| 13 | 12 | sneqd 4638 |
. . . . . . . 8
⊢ (𝑝 = 𝑋 → {〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉} = {〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉}) |
| 14 | 13 | ssiun2s 5048 |
. . . . . . 7
⊢ (𝑋 ∈ 𝑉 → {〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉} ⊆ ∪ 𝑝 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉}) |
| 15 | 14 | 3ad2ant2 1135 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉} ⊆ ∪ 𝑝 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉}) |
| 16 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑞 = 𝑌 → (𝐹‘𝑞) = (𝐹‘𝑌)) |
| 17 | 16 | opeq2d 4880 |
. . . . . . . . . . . 12
⊢ (𝑞 = 𝑌 → 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 = 〈(𝐹‘𝑝), (𝐹‘𝑌)〉) |
| 18 | | oveq2 7439 |
. . . . . . . . . . . . 13
⊢ (𝑞 = 𝑌 → (𝑝 · 𝑞) = (𝑝 · 𝑌)) |
| 19 | 18 | fveq2d 6910 |
. . . . . . . . . . . 12
⊢ (𝑞 = 𝑌 → (𝐹‘(𝑝 · 𝑞)) = (𝐹‘(𝑝 · 𝑌))) |
| 20 | 17, 19 | opeq12d 4881 |
. . . . . . . . . . 11
⊢ (𝑞 = 𝑌 → 〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉 = 〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉) |
| 21 | 20 | sneqd 4638 |
. . . . . . . . . 10
⊢ (𝑞 = 𝑌 → {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} = {〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉}) |
| 22 | 21 | ssiun2s 5048 |
. . . . . . . . 9
⊢ (𝑌 ∈ 𝑉 → {〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉} ⊆ ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
| 23 | 22 | ralrimivw 3150 |
. . . . . . . 8
⊢ (𝑌 ∈ 𝑉 → ∀𝑝 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑌)〉, (𝐹‘(𝑝 · 𝑌))〉} ⊆ ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
| 24 | | ss2iun 5010 |
. . . . . . . 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))〉}) |
| 25 | 23, 24 | 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))〉}) |
| 26 | 25 | 3ad2ant3 1136 |
. . . . . 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))〉}) |
| 27 | 15, 26 | sstrd 3994 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉} ⊆ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
| 28 | 4 | 3ad2ant1 1134 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
| 29 | 27, 28 | sseqtrrd 4021 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → {〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉} ⊆ ∙ ) |
| 30 | | opex 5469 |
. . . . 5
⊢
〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉 ∈ V |
| 31 | 30 | snss 4785 |
. . . 4
⊢
(〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉 ∈ ∙ ↔
{〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉} ⊆ ∙ ) |
| 32 | 29, 31 | sylibr 234 |
. . 3
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → 〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉 ∈ ∙ ) |
| 33 | | funopfv 6958 |
. . 3
⊢ (Fun
∙
→ (〈〈(𝐹‘𝑋), (𝐹‘𝑌)〉, (𝐹‘(𝑋 · 𝑌))〉 ∈ ∙ → ( ∙
‘〈(𝐹‘𝑋), (𝐹‘𝑌)〉) = (𝐹‘(𝑋 · 𝑌)))) |
| 34 | 8, 32, 33 | sylc 65 |
. 2
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ( ∙ ‘〈(𝐹‘𝑋), (𝐹‘𝑌)〉) = (𝐹‘(𝑋 · 𝑌))) |
| 35 | 1, 34 | eqtrid 2789 |
1
⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ((𝐹‘𝑋) ∙ (𝐹‘𝑌)) = (𝐹‘(𝑋 · 𝑌))) |