Theorem assintopcllaw 49005

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 6916	. 2 ⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → 𝑀 ∈ V)
2		assintopval 48998	. . . . 5 ⊢ (𝑀 ∈ V → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
3	2	eleq2d 2848	. . . 4 ⊢ (𝑀 ∈ V → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ⚬ ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀}))
4		breq1 5111	. . . . 5 ⊢ (𝑜 = ⚬ → (𝑜 assLaw 𝑀 ↔ ⚬ assLaw 𝑀))
5	4	elrab 3649	. . . 4 ⊢ ( ⚬ ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ↔ ( ⚬ ∈ ( clIntOp ‘𝑀) ∧ ⚬ assLaw 𝑀))
6	3, 5	bitrdi 290	. . 3 ⊢ (𝑀 ∈ V → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ( ⚬ ∈ ( clIntOp ‘𝑀) ∧ ⚬ assLaw 𝑀)))
7		clintopcllaw 49004	. . . 4 ⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ clLaw 𝑀)
8	7	adantr 485	. . 3 ⊢ (( ⚬ ∈ ( clIntOp ‘𝑀) ∧ ⚬ assLaw 𝑀) → ⚬ clLaw 𝑀)
9	6, 8	biimtrdi 256	. 2 ⊢ (𝑀 ∈ V → ( ⚬ ∈ ( assIntOp ‘𝑀) → ⚬ clLaw 𝑀))
10	1, 9	mpcom 39	1 ⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ⚬ clLaw 𝑀)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∈ wcel 2142  {crab 3415  Vcvv 3454   class class class wbr 5108  ‘cfv 6536   clLaw ccllaw 48976   assLaw casslaw 48977   clIntOp cclintop 48990   assIntOp cassintop 48991

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8824  df-cllaw 48979  df-intop 48992  df-clintop 48993  df-assintop 48994

This theorem is used by: (None)