Theorem assintopcllaw 47187

Description: The closure low holds for an associative (closed internal binary) operation for a set. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)

Assertion

Ref	Expression
assintopcllaw	⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ⚬ clLaw 𝑀)

Proof of Theorem assintopcllaw

Dummy variable 𝑜 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6929	. 2 ⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → 𝑀 ∈ V)
2		assintopval 47180	. . . . 5 ⊢ (𝑀 ∈ V → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
3	2	eleq2d 2814	. . . 4 ⊢ (𝑀 ∈ V → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ⚬ ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀}))
4		breq1 5145	. . . . 5 ⊢ (𝑜 = ⚬ → (𝑜 assLaw 𝑀 ↔ ⚬ assLaw 𝑀))
5	4	elrab 3680	. . . 4 ⊢ ( ⚬ ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ↔ ( ⚬ ∈ ( clIntOp ‘𝑀) ∧ ⚬ assLaw 𝑀))
6	3, 5	bitrdi 287	. . 3 ⊢ (𝑀 ∈ V → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ( ⚬ ∈ ( clIntOp ‘𝑀) ∧ ⚬ assLaw 𝑀)))
7		clintopcllaw 47186	. . . 4 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ clLaw 𝑀)
8	7	adantr 480	. . 3 ⊢ (( ⚬ ∈ ( clIntOp ‘𝑀) ∧ ⚬ assLaw 𝑀) → ⚬ clLaw 𝑀)
9	6, 8	syl6bi 253	. 2 ⊢ (𝑀 ∈ V → ( ⚬ ∈ ( assIntOp ‘𝑀) → ⚬ clLaw 𝑀))
10	1, 9	mpcom 38	1 ⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ⚬ clLaw 𝑀)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2099  {crab 3427  Vcvv 3469   class class class wbr 5142  ‘cfv 6542   clLaw ccllaw 47158   assLaw casslaw 47159   clIntOp cclintop 47172   assIntOp cassintop 47173

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8836  df-cllaw 47161  df-intop 47174  df-clintop 47175  df-assintop 47176

This theorem is referenced by: (None)