Theorem caovassd 6171

Description: Convert an operation associative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014.)

Hypotheses

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)

Assertion

Ref	Expression
caovassd	⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem caovassd

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 6170	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
7	1, 2, 3, 4, 6	syl13anc 1273	1 ⊢ (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  (class class class)co 6007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6010

This theorem is referenced by:  caov32d  6192  caov12d  6193  caov13d  6195  caov4d  6196  caovdilemd  6203  caovimo  6205  enq0tr  7632  prarloclemlo  7692  ltsosr  7962  seqf1oglem2a  10752  grpinvalem  13433  grpinva  13434  grprida  13435  grprcan  13585