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Mirrors > Home > ILE Home > Th. List > caovassd | 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 19 | . 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 5817 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) |
7 | 1, 2, 3, 4, 6 | syl13anc 1177 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 925 = wceq 1290 ∈ wcel 1439 (class class class)co 5666 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-un 3004 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-iota 4993 df-fv 5036 df-ov 5669 |
This theorem is referenced by: caov32d 5839 caov12d 5840 caov13d 5842 caov4d 5843 caovdilemd 5850 caovimo 5852 grprinvlem 5853 grprinvd 5854 grpridd 5855 enq0tr 7054 prarloclemlo 7114 ltsosr 7371 |
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