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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 7464 | . . 3 ⊢ (𝐴𝐺𝐶) ∈ V | |
| 2 | ovex 7464 | . . 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 7667 | . 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 7633 | . . 3 ⊢ ((𝐴𝐺𝐶)𝐺𝐻) = (𝐴𝐺(𝐶𝐺𝐻)) | 
| 11 | caovdir.2 | . . . 4 ⊢ 𝐵 ∈ V | |
| 12 | caovdl.4 | . . . 4 ⊢ 𝐷 ∈ V | |
| 13 | 11, 12, 3, 9 | caovass 7633 | . . 3 ⊢ ((𝐵𝐺𝐷)𝐺𝐻) = (𝐵𝐺(𝐷𝐺𝐻)) | 
| 14 | 10, 13 | oveq12i 7443 | . 2 ⊢ (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻)) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))) | 
| 15 | 6, 14 | eqtri 2765 | 1 ⊢ (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 (class class class)co 7431 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 | 
| This theorem is referenced by: caovlem2 7669 axmulass 11197 | 
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