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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 7335 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) |
7 | 1, 2, 3, 4, 6 | syl13anc 1364 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 (class class class)co 7145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 |
This theorem is referenced by: caov32d 7357 caov12d 7358 caov13d 7360 caov4d 7361 seqf1olem2a 13396 grprinvlem 17871 grprinvd 17872 grpridd 17873 grprcan 18075 |
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