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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 2203 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
2 | 1 | anbi1d 460 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
3 | 2 | opabbidv 3994 | . 2 ⊢ (𝐴 = 𝐵 → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}) |
4 | df-xp 4545 | . 2 ⊢ (𝐴 × 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶)} | |
5 | df-xp 4545 | . 2 ⊢ (𝐵 × 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} | |
6 | 3, 4, 5 | 3eqtr4g 2197 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 {copab 3988 × cxp 4537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-opab 3990 df-xp 4545 |
This theorem is referenced by: xpeq12 4558 xpeq1i 4559 xpeq1d 4562 opthprc 4590 reseq2 4814 xpeq0r 4961 xpdisj1 4963 xpima1 4985 pmvalg 6553 xpsneng 6716 xpcomeng 6722 xpdom2g 6726 xpfi 6818 exmidomni 7014 exmidfodomrlemim 7057 hashxp 10572 txuni2 12425 txbas 12427 txopn 12434 txrest 12445 txdis 12446 txdis1cn 12447 xmettxlem 12678 xmettx 12679 |
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