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Mirrors > Home > ILE Home > Th. List > caovdir2d | GIF version |
Description: Convert an operation distributive law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.) |
Ref | Expression |
---|---|
caovdir2d.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))) |
caovdir2d.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
caovdir2d.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑆) |
caovdir2d.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑆) |
caovdir2d.cl | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) |
caovdir2d.com | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥)) |
Ref | Expression |
---|---|
caovdir2d | ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovdir2d.1 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))) | |
2 | caovdir2d.4 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
3 | caovdir2d.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
4 | caovdir2d.3 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
5 | 1, 2, 3, 4 | caovdid 5996 | . 2 ⊢ (𝜑 → (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵))) |
6 | caovdir2d.com | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥)) | |
7 | caovdir2d.cl | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆) | |
8 | 7, 3, 4 | caovcld 5974 | . . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝑆) |
9 | 6, 8, 2 | caovcomd 5977 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = (𝐶𝐺(𝐴𝐹𝐵))) |
10 | 6, 3, 2 | caovcomd 5977 | . . 3 ⊢ (𝜑 → (𝐴𝐺𝐶) = (𝐶𝐺𝐴)) |
11 | 6, 4, 2 | caovcomd 5977 | . . 3 ⊢ (𝜑 → (𝐵𝐺𝐶) = (𝐶𝐺𝐵)) |
12 | 10, 11 | oveq12d 5842 | . 2 ⊢ (𝜑 → ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵))) |
13 | 5, 9, 12 | 3eqtr4d 2200 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 963 = wceq 1335 ∈ wcel 2128 (class class class)co 5824 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-iota 5135 df-fv 5178 df-ov 5827 |
This theorem is referenced by: addcmpblnq 7287 ltanqg 7320 addcmpblnq0 7363 mulasssrg 7678 mulgt0sr 7698 mulextsr1lem 7700 |
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