Theorem assintopcllaw 48922

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 48915	. . . . 5 ⊢ (𝑀 ∈ V → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
3	2	eleq2d 2847	. . . 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 48921	. . . 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)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2141  {crab 3414  Vcvv 3453   class class class wbr 5108  ‘cfv 6536   clLaw ccllaw 48893   assLaw casslaw 48894   clIntOp cclintop 48907   assIntOp cassintop 48908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  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 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8825  df-cllaw 48896  df-intop 48909  df-clintop 48910  df-assintop 48911

This theorem is referenced by: (None)