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Mirrors > Home > MPE Home > Th. List > caovdilem | Structured version Visualization version GIF version |
Description: Lemma used by real number construction. (Contributed by NM, 26-Aug-1995.) |
Ref | Expression |
---|---|
caovdir.1 | ⊢ 𝐴 ∈ V |
caovdir.2 | ⊢ 𝐵 ∈ V |
caovdir.3 | ⊢ 𝐶 ∈ V |
caovdir.com | ⊢ (𝑥𝐺𝑦) = (𝑦𝐺𝑥) |
caovdir.distr | ⊢ (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)) |
caovdl.4 | ⊢ 𝐷 ∈ V |
caovdl.5 | ⊢ 𝐻 ∈ V |
caovdl.ass | ⊢ ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) |
Ref | Expression |
---|---|
caovdilem | ⊢ (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7394 | . . 3 ⊢ (𝐴𝐺𝐶) ∈ V | |
2 | ovex 7394 | . . 3 ⊢ (𝐵𝐺𝐷) ∈ V | |
3 | caovdl.5 | . . 3 ⊢ 𝐻 ∈ V | |
4 | caovdir.com | . . 3 ⊢ (𝑥𝐺𝑦) = (𝑦𝐺𝑥) | |
5 | caovdir.distr | . . 3 ⊢ (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧)) | |
6 | 1, 2, 3, 4, 5 | caovdir 7592 | . 2 ⊢ (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻)) |
7 | caovdir.1 | . . . 4 ⊢ 𝐴 ∈ V | |
8 | caovdir.3 | . . . 4 ⊢ 𝐶 ∈ V | |
9 | caovdl.ass | . . . 4 ⊢ ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)) | |
10 | 7, 8, 3, 9 | caovass 7558 | . . 3 ⊢ ((𝐴𝐺𝐶)𝐺𝐻) = (𝐴𝐺(𝐶𝐺𝐻)) |
11 | caovdir.2 | . . . 4 ⊢ 𝐵 ∈ V | |
12 | caovdl.4 | . . . 4 ⊢ 𝐷 ∈ V | |
13 | 11, 12, 3, 9 | caovass 7558 | . . 3 ⊢ ((𝐵𝐺𝐷)𝐺𝐻) = (𝐵𝐺(𝐷𝐺𝐻)) |
14 | 10, 13 | oveq12i 7373 | . 2 ⊢ (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻)) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))) |
15 | 6, 14 | eqtri 2761 | 1 ⊢ (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ 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 ax-nul 5267 |
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-ne 2941 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: caovlem2 7594 axmulass 11101 |
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