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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 5676 | . 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 5654 | . . 3 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝑆) |
9 | 6, 8, 2 | caovcomd 5657 | . 2 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = (𝐶𝐺(𝐴𝐹𝐵))) |
10 | 6, 3, 2 | caovcomd 5657 | . . 3 ⊢ (𝜑 → (𝐴𝐺𝐶) = (𝐶𝐺𝐴)) |
11 | 6, 4, 2 | caovcomd 5657 | . . 3 ⊢ (𝜑 → (𝐵𝐺𝐶) = (𝐶𝐺𝐵)) |
12 | 10, 11 | oveq12d 5530 | . 2 ⊢ (𝜑 → ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵))) |
13 | 5, 9, 12 | 3eqtr4d 2082 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∧ w3a 885 = wceq 1243 ∈ wcel 1393 (class class class)co 5512 |
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-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
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-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-iota 4867 df-fv 4910 df-ov 5515 |
This theorem is referenced by: addcmpblnq 6465 ltanqg 6498 addcmpblnq0 6541 mulasssrg 6843 mulgt0sr 6862 mulextsr1lem 6864 |
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