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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 1487 | . . 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 6848 | . . 3 ⊢ ((⊤ ∧ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶))) |
8 | 4, 7 | mpan 706 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶))) |
9 | 1, 2, 3, 8 | mp3an 1424 | 1 ⊢ (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 ∧ w3a 1037 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 Vcvv 3200 (class class class)co 6650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: caovdir 6868 caovlem2 6870 |
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