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Mirrors > Home > ILE Home > Th. List > caov42d | GIF version |
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.) |
Ref | Expression |
---|---|
caovd.1 | ⊢ (φ → A ∈ 𝑆) |
caovd.2 | ⊢ (φ → B ∈ 𝑆) |
caovd.3 | ⊢ (φ → 𝐶 ∈ 𝑆) |
caovd.com | ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x𝐹y) = (y𝐹x)) |
caovd.ass | ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z))) |
caovd.4 | ⊢ (φ → 𝐷 ∈ 𝑆) |
caovd.cl | ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x𝐹y) ∈ 𝑆) |
Ref | Expression |
---|---|
caov42d | ⊢ (φ → ((A𝐹B)𝐹(𝐶𝐹𝐷)) = ((A𝐹𝐶)𝐹(𝐷𝐹B))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovd.1 | . . 3 ⊢ (φ → A ∈ 𝑆) | |
2 | caovd.2 | . . 3 ⊢ (φ → B ∈ 𝑆) | |
3 | caovd.3 | . . 3 ⊢ (φ → 𝐶 ∈ 𝑆) | |
4 | caovd.com | . . 3 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x𝐹y) = (y𝐹x)) | |
5 | caovd.ass | . . 3 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆 ∧ z ∈ 𝑆)) → ((x𝐹y)𝐹z) = (x𝐹(y𝐹z))) | |
6 | caovd.4 | . . 3 ⊢ (φ → 𝐷 ∈ 𝑆) | |
7 | caovd.cl | . . 3 ⊢ ((φ ∧ (x ∈ 𝑆 ∧ y ∈ 𝑆)) → (x𝐹y) ∈ 𝑆) | |
8 | 1, 2, 3, 4, 5, 6, 7 | caov4d 5627 | . 2 ⊢ (φ → ((A𝐹B)𝐹(𝐶𝐹𝐷)) = ((A𝐹𝐶)𝐹(B𝐹𝐷))) |
9 | 4, 2, 6 | caovcomd 5599 | . . 3 ⊢ (φ → (B𝐹𝐷) = (𝐷𝐹B)) |
10 | 9 | oveq2d 5471 | . 2 ⊢ (φ → ((A𝐹𝐶)𝐹(B𝐹𝐷)) = ((A𝐹𝐶)𝐹(𝐷𝐹B))) |
11 | 8, 10 | eqtrd 2069 | 1 ⊢ (φ → ((A𝐹B)𝐹(𝐶𝐹𝐷)) = ((A𝐹𝐶)𝐹(𝐷𝐹B))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∧ w3a 884 = wceq 1242 ∈ wcel 1390 (class class class)co 5455 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-iota 4810 df-fv 4853 df-ov 5458 |
This theorem is referenced by: caovlem2d 5635 mulcmpblnrlemg 6668 ltasrg 6698 axmulass 6757 |
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