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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 6678 | . . 3 ⊢ (𝐴𝐺𝐶) ∈ V | |
2 | ovex 6678 | . . 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 6868 | . 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 6834 | . . 3 ⊢ ((𝐴𝐺𝐶)𝐺𝐻) = (𝐴𝐺(𝐶𝐺𝐻)) |
11 | caovdir.2 | . . . 4 ⊢ 𝐵 ∈ V | |
12 | caovdl.4 | . . . 4 ⊢ 𝐷 ∈ V | |
13 | 11, 12, 3, 9 | caovass 6834 | . . 3 ⊢ ((𝐵𝐺𝐷)𝐺𝐻) = (𝐵𝐺(𝐷𝐺𝐻)) |
14 | 10, 13 | oveq12i 6662 | . 2 ⊢ (((𝐴𝐺𝐶)𝐺𝐻)𝐹((𝐵𝐺𝐷)𝐺𝐻)) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))) |
15 | 6, 14 | eqtri 2644 | 1 ⊢ (((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐷))𝐺𝐻) = ((𝐴𝐺(𝐶𝐺𝐻))𝐹(𝐵𝐺(𝐷𝐺𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ 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 ax-nul 4789 |
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-eu 2474 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-sbc 3436 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: caovlem2 6870 axmulass 9978 |
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