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Mirrors > Home > ILE Home > Th. List > xpeq1 | GIF version |
Description: Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.) |
Ref | Expression |
---|---|
xpeq1 | ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2101 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
2 | 1 | anbi1d 438 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
3 | 2 | opabbidv 3823 | . 2 ⊢ (𝐴 = 𝐵 → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}) |
4 | df-xp 4351 | . 2 ⊢ (𝐴 × 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} | |
5 | df-xp 4351 | . 2 ⊢ (𝐵 × 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} | |
6 | 3, 4, 5 | 3eqtr4g 2097 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 {copab 3817 × cxp 4343 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-opab 3819 df-xp 4351 |
This theorem is referenced by: xpeq12 4364 xpeq1i 4365 xpeq1d 4368 opthprc 4391 reseq2 4607 xpeq0r 4746 xpdisj1 4747 xpima1 4767 xpsneng 6296 xpcomeng 6302 xpdom2g 6306 |
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