Theorem assintopass 47169

Description: An associative (closed internal binary) operation for a set is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)

Assertion

Ref	Expression
assintopass	⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))

Distinct variable groups:   𝑥,𝑀,𝑦,𝑧   𝑥, ⚬ ,𝑦,𝑧

Proof of Theorem assintopass

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ⚬ ∈ ( assIntOp ‘𝑀))
2		elfvex 6923	. 2 ⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → 𝑀 ∈ V)
3		assintopasslaw 47168	. . 3 ⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ⚬ assLaw 𝑀)
4		isasslaw 47147	. . 3 ⊢ (( ⚬ ∈ ( assIntOp ‘𝑀) ∧ 𝑀 ∈ V) → ( ⚬ assLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
5	3, 4	syl5ibcom 244	. 2 ⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → (( ⚬ ∈ ( assIntOp ‘𝑀) ∧ 𝑀 ∈ V) → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
6	1, 2, 5	mp2and 696	1 ⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3055  Vcvv 3468   class class class wbr 5141  ‘cfv 6537  (class class class)co 7405   assLaw casslaw 47139   assIntOp cassintop 47153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-map 8824  df-asslaw 47143  df-intop 47154  df-clintop 47155  df-assintop 47156

This theorem is referenced by: (None)