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Mirrors > Home > ILE Home > Th. List > ecovicom | GIF version |
Description: Lemma used to transfer a commutative law via an equivalence relation. (Contributed by Jim Kingdon, 15-Sep-2019.) |
Ref | Expression |
---|---|
ecovicom.1 | ⊢ 𝐶 = ((𝑆 × 𝑆) / ∼ ) |
ecovicom.2 | ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ + [〈𝑧, 𝑤〉] ∼ ) = [〈𝐷, 𝐺〉] ∼ ) |
ecovicom.3 | ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ([〈𝑧, 𝑤〉] ∼ + [〈𝑥, 𝑦〉] ∼ ) = [〈𝐻, 𝐽〉] ∼ ) |
ecovicom.4 | ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝐷 = 𝐻) |
ecovicom.5 | ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝐺 = 𝐽) |
Ref | Expression |
---|---|
ecovicom | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecovicom.1 | . 2 ⊢ 𝐶 = ((𝑆 × 𝑆) / ∼ ) | |
2 | oveq1 5675 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ∼ = 𝐴 → ([〈𝑥, 𝑦〉] ∼ + [〈𝑧, 𝑤〉] ∼ ) = (𝐴 + [〈𝑧, 𝑤〉] ∼ )) | |
3 | oveq2 5676 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ∼ = 𝐴 → ([〈𝑧, 𝑤〉] ∼ + [〈𝑥, 𝑦〉] ∼ ) = ([〈𝑧, 𝑤〉] ∼ + 𝐴)) | |
4 | 2, 3 | eqeq12d 2103 | . 2 ⊢ ([〈𝑥, 𝑦〉] ∼ = 𝐴 → (([〈𝑥, 𝑦〉] ∼ + [〈𝑧, 𝑤〉] ∼ ) = ([〈𝑧, 𝑤〉] ∼ + [〈𝑥, 𝑦〉] ∼ ) ↔ (𝐴 + [〈𝑧, 𝑤〉] ∼ ) = ([〈𝑧, 𝑤〉] ∼ + 𝐴))) |
5 | oveq2 5676 | . . 3 ⊢ ([〈𝑧, 𝑤〉] ∼ = 𝐵 → (𝐴 + [〈𝑧, 𝑤〉] ∼ ) = (𝐴 + 𝐵)) | |
6 | oveq1 5675 | . . 3 ⊢ ([〈𝑧, 𝑤〉] ∼ = 𝐵 → ([〈𝑧, 𝑤〉] ∼ + 𝐴) = (𝐵 + 𝐴)) | |
7 | 5, 6 | eqeq12d 2103 | . 2 ⊢ ([〈𝑧, 𝑤〉] ∼ = 𝐵 → ((𝐴 + [〈𝑧, 𝑤〉] ∼ ) = ([〈𝑧, 𝑤〉] ∼ + 𝐴) ↔ (𝐴 + 𝐵) = (𝐵 + 𝐴))) |
8 | ecovicom.4 | . . . 4 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝐷 = 𝐻) | |
9 | ecovicom.5 | . . . 4 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → 𝐺 = 𝐽) | |
10 | opeq12 3632 | . . . . 5 ⊢ ((𝐷 = 𝐻 ∧ 𝐺 = 𝐽) → 〈𝐷, 𝐺〉 = 〈𝐻, 𝐽〉) | |
11 | 10 | eceq1d 6344 | . . . 4 ⊢ ((𝐷 = 𝐻 ∧ 𝐺 = 𝐽) → [〈𝐷, 𝐺〉] ∼ = [〈𝐻, 𝐽〉] ∼ ) |
12 | 8, 9, 11 | syl2anc 404 | . . 3 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → [〈𝐷, 𝐺〉] ∼ = [〈𝐻, 𝐽〉] ∼ ) |
13 | ecovicom.2 | . . 3 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ + [〈𝑧, 𝑤〉] ∼ ) = [〈𝐷, 𝐺〉] ∼ ) | |
14 | ecovicom.3 | . . . 4 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ([〈𝑧, 𝑤〉] ∼ + [〈𝑥, 𝑦〉] ∼ ) = [〈𝐻, 𝐽〉] ∼ ) | |
15 | 14 | ancoms 265 | . . 3 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([〈𝑧, 𝑤〉] ∼ + [〈𝑥, 𝑦〉] ∼ ) = [〈𝐻, 𝐽〉] ∼ ) |
16 | 12, 13, 15 | 3eqtr4d 2131 | . 2 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ + [〈𝑧, 𝑤〉] ∼ ) = ([〈𝑧, 𝑤〉] ∼ + [〈𝑥, 𝑦〉] ∼ )) |
17 | 1, 4, 7, 16 | 2ecoptocl 6396 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1290 ∈ wcel 1439 〈cop 3455 × cxp 4452 (class class class)co 5668 [cec 6306 / cqs 6307 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2624 df-un 3006 df-in 3008 df-ss 3015 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-br 3854 df-opab 3908 df-xp 4460 df-cnv 4462 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-iota 4995 df-fv 5038 df-ov 5671 df-ec 6310 df-qs 6314 |
This theorem is referenced by: addcomnqg 7003 mulcomnqg 7005 addcomsrg 7364 mulcomsrg 7366 axmulcom 7469 |
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