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Mirrors > Home > MPE Home > Th. List > caovdi | Structured version Visualization version GIF version |
Description: Convert an operation distributive law to class notation. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 28-Jun-2013.) |
Ref | Expression |
---|---|
caovdi.1 | ⊢ 𝐴 ∈ V |
caovdi.2 | ⊢ 𝐵 ∈ V |
caovdi.3 | ⊢ 𝐶 ∈ V |
caovdi.4 | ⊢ (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)) |
Ref | Expression |
---|---|
caovdi | ⊢ (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovdi.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | caovdi.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | caovdi.3 | . 2 ⊢ 𝐶 ∈ V | |
4 | tru 1546 | . . 3 ⊢ ⊤ | |
5 | caovdi.4 | . . . . 5 ⊢ (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)) | |
6 | 5 | a1i 11 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V ∧ 𝑧 ∈ V)) → (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))) |
7 | 6 | caovdig 7572 | . . 3 ⊢ ((⊤ ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶))) |
8 | 4, 7 | mpan 689 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶))) |
9 | 1, 2, 3, 8 | mp3an 1462 | 1 ⊢ (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∧ w3a 1088 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 Vcvv 3447 (class class class)co 7361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-iota 6452 df-fv 6508 df-ov 7364 |
This theorem is referenced by: caovdir 7592 caovlem2 7594 |
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