Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ecovcom | GIF version |
Description: Lemma used to transfer a commutative law via an equivalence relation. Most uses will want ecovicom 6214 instead. (Contributed by NM, 29-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.) |
Ref | Expression |
---|---|
ecovcom.1 | ⊢ 𝐶 = ((𝑆 × 𝑆) / ∼ ) |
ecovcom.2 | ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ + [〈𝑧, 𝑤〉] ∼ ) = [〈𝐷, 𝐺〉] ∼ ) |
ecovcom.3 | ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ([〈𝑧, 𝑤〉] ∼ + [〈𝑥, 𝑦〉] ∼ ) = [〈𝐻, 𝐽〉] ∼ ) |
ecovcom.4 | ⊢ 𝐷 = 𝐻 |
ecovcom.5 | ⊢ 𝐺 = 𝐽 |
Ref | Expression |
---|---|
ecovcom | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecovcom.1 | . 2 ⊢ 𝐶 = ((𝑆 × 𝑆) / ∼ ) | |
2 | oveq1 5519 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ∼ = 𝐴 → ([〈𝑥, 𝑦〉] ∼ + [〈𝑧, 𝑤〉] ∼ ) = (𝐴 + [〈𝑧, 𝑤〉] ∼ )) | |
3 | oveq2 5520 | . . 3 ⊢ ([〈𝑥, 𝑦〉] ∼ = 𝐴 → ([〈𝑧, 𝑤〉] ∼ + [〈𝑥, 𝑦〉] ∼ ) = ([〈𝑧, 𝑤〉] ∼ + 𝐴)) | |
4 | 2, 3 | eqeq12d 2054 | . 2 ⊢ ([〈𝑥, 𝑦〉] ∼ = 𝐴 → (([〈𝑥, 𝑦〉] ∼ + [〈𝑧, 𝑤〉] ∼ ) = ([〈𝑧, 𝑤〉] ∼ + [〈𝑥, 𝑦〉] ∼ ) ↔ (𝐴 + [〈𝑧, 𝑤〉] ∼ ) = ([〈𝑧, 𝑤〉] ∼ + 𝐴))) |
5 | oveq2 5520 | . . 3 ⊢ ([〈𝑧, 𝑤〉] ∼ = 𝐵 → (𝐴 + [〈𝑧, 𝑤〉] ∼ ) = (𝐴 + 𝐵)) | |
6 | oveq1 5519 | . . 3 ⊢ ([〈𝑧, 𝑤〉] ∼ = 𝐵 → ([〈𝑧, 𝑤〉] ∼ + 𝐴) = (𝐵 + 𝐴)) | |
7 | 5, 6 | eqeq12d 2054 | . 2 ⊢ ([〈𝑧, 𝑤〉] ∼ = 𝐵 → ((𝐴 + [〈𝑧, 𝑤〉] ∼ ) = ([〈𝑧, 𝑤〉] ∼ + 𝐴) ↔ (𝐴 + 𝐵) = (𝐵 + 𝐴))) |
8 | ecovcom.4 | . . . 4 ⊢ 𝐷 = 𝐻 | |
9 | ecovcom.5 | . . . 4 ⊢ 𝐺 = 𝐽 | |
10 | opeq12 3551 | . . . . 5 ⊢ ((𝐷 = 𝐻 ∧ 𝐺 = 𝐽) → 〈𝐷, 𝐺〉 = 〈𝐻, 𝐽〉) | |
11 | 10 | eceq1d 6142 | . . . 4 ⊢ ((𝐷 = 𝐻 ∧ 𝐺 = 𝐽) → [〈𝐷, 𝐺〉] ∼ = [〈𝐻, 𝐽〉] ∼ ) |
12 | 8, 9, 11 | mp2an 402 | . . 3 ⊢ [〈𝐷, 𝐺〉] ∼ = [〈𝐻, 𝐽〉] ∼ |
13 | ecovcom.2 | . . 3 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ + [〈𝑧, 𝑤〉] ∼ ) = [〈𝐷, 𝐺〉] ∼ ) | |
14 | ecovcom.3 | . . . 4 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ([〈𝑧, 𝑤〉] ∼ + [〈𝑥, 𝑦〉] ∼ ) = [〈𝐻, 𝐽〉] ∼ ) | |
15 | 14 | ancoms 255 | . . 3 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([〈𝑧, 𝑤〉] ∼ + [〈𝑥, 𝑦〉] ∼ ) = [〈𝐻, 𝐽〉] ∼ ) |
16 | 12, 13, 15 | 3eqtr4a 2098 | . 2 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([〈𝑥, 𝑦〉] ∼ + [〈𝑧, 𝑤〉] ∼ ) = ([〈𝑧, 𝑤〉] ∼ + [〈𝑥, 𝑦〉] ∼ )) |
17 | 1, 4, 7, 16 | 2ecoptocl 6194 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 + 𝐵) = (𝐵 + 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 〈cop 3378 × cxp 4343 (class class class)co 5512 [cec 6104 / cqs 6105 |
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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fv 4910 df-ov 5515 df-ec 6108 df-qs 6112 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |