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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 5789 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ∼ = 𝐴 → ([〈𝑥, 𝑦〉] ∼ + [〈𝑧, 𝑤〉] ∼ ) = (𝐴 + [〈𝑧, 𝑤〉] ∼ )) | |
3 | oveq2 5790 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ∼ = 𝐴 → ([〈𝑧, 𝑤〉] ∼ + [〈𝑥, 𝑦〉] ∼ ) = ([〈𝑧, 𝑤〉] ∼ + 𝐴)) | |
4 | 2, 3 | eqeq12d 2155 | . 2 ⊢ ([〈𝑥, 𝑦〉] ∼ = 𝐴 → (([〈𝑥, 𝑦〉] ∼ + [〈𝑧, 𝑤〉] ∼ ) = ([〈𝑧, 𝑤〉] ∼ + [〈𝑥, 𝑦〉] ∼ ) ↔ (𝐴 + [〈𝑧, 𝑤〉] ∼ ) = ([〈𝑧, 𝑤〉] ∼ + 𝐴))) |
5 | oveq2 5790 | . . 3 ⊢ ([〈𝑧, 𝑤〉] ∼ = 𝐵 → (𝐴 + [〈𝑧, 𝑤〉] ∼ ) = (𝐴 + 𝐵)) | |
6 | oveq1 5789 | . . 3 ⊢ ([〈𝑧, 𝑤〉] ∼ = 𝐵 → ([〈𝑧, 𝑤〉] ∼ + 𝐴) = (𝐵 + 𝐴)) | |
7 | 5, 6 | eqeq12d 2155 | . 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 3715 | . . . . 5 ⊢ ((𝐷 = 𝐻 ∧ 𝐺 = 𝐽) → 〈𝐷, 𝐺〉 = 〈𝐻, 𝐽〉) | |
11 | 10 | eceq1d 6473 | . . . 4 ⊢ ((𝐷 = 𝐻 ∧ 𝐺 = 𝐽) → [〈𝐷, 𝐺〉] ∼ = [〈𝐻, 𝐽〉] ∼ ) |
12 | 8, 9, 11 | syl2anc 409 | . . 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 266 | . . 3 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([〈𝑧, 𝑤〉] ∼ + [〈𝑥, 𝑦〉] ∼ ) = [〈𝐻, 𝐽〉] ∼ ) |
16 | 12, 13, 15 | 3eqtr4d 2183 | . 2 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ + [〈𝑧, 𝑤〉] ∼ ) = ([〈𝑧, 𝑤〉] ∼ + [〈𝑥, 𝑦〉] ∼ )) |
17 | 1, 4, 7, 16 | 2ecoptocl 6525 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 〈cop 3535 × cxp 4545 (class class class)co 5782 [cec 6435 / cqs 6436 |
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-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fv 5139 df-ov 5785 df-ec 6439 df-qs 6443 |
This theorem is referenced by: addcomnqg 7213 mulcomnqg 7215 addcomsrg 7587 mulcomsrg 7589 axmulcom 7703 |
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