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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 7191 | . . 3 ⊢ (𝐴𝐺𝐶) ∈ V | |
2 | ovex 7191 | . . 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 7384 | . 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 7350 | . . 3 ⊢ ((𝐴𝐺𝐶)𝐺𝐻) = (𝐴𝐺(𝐶𝐺𝐻)) |
11 | caovdir.2 | . . . 4 ⊢ 𝐵 ∈ V | |
12 | caovdl.4 | . . . 4 ⊢ 𝐷 ∈ V | |
13 | 11, 12, 3, 9 | caovass 7350 | . . 3 ⊢ ((𝐵𝐺𝐷)𝐺𝐻) = (𝐵𝐺(𝐷𝐺𝐻)) |
14 | 10, 13 | oveq12i 7170 | . 2 ⊢ (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻)) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))) |
15 | 6, 14 | eqtri 2846 | 1 ⊢ (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3496 (class class class)co 7158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 |
This theorem is referenced by: caovlem2 7386 axmulass 10581 |
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