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| Mirrors > Home > MPE Home > Th. List > caovassd | Structured version Visualization version GIF version | ||
| Description: Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.) | 
| Ref | Expression | 
|---|---|
| caovassg.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) | 
| caovassd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) | 
| caovassd.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑆) | 
| caovassd.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑆) | 
| Ref | Expression | 
|---|---|
| caovassd | ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | id 22 | . 2 ⊢ (𝜑 → 𝜑) | |
| 2 | caovassd.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
| 3 | caovassd.3 | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
| 4 | caovassd.4 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
| 5 | caovassg.1 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))) | |
| 6 | 5 | caovassg 7631 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) | 
| 7 | 1, 2, 3, 4, 6 | syl13anc 1374 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 (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 | 
| 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-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: caov32d 7653 caov12d 7654 caov13d 7656 caov4d 7657 seqf1olem2a 14081 grpinvalem 18686 grpinva 18687 grprida 18688 grprcan 18991 | 
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