Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clintopcllaw Structured version   Visualization version   GIF version

Theorem clintopcllaw 44103
 Description: The closure law holds for a closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
Assertion
Ref Expression
clintopcllaw ( ∈ ( clIntOp ‘𝑀) → clLaw 𝑀)

Proof of Theorem clintopcllaw
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clintop 44100 . . 3 ( ∈ ( clIntOp ‘𝑀) → :(𝑀 × 𝑀)⟶𝑀)
2 ffnov 7270 . . . 4 ( :(𝑀 × 𝑀)⟶𝑀 ↔ ( Fn (𝑀 × 𝑀) ∧ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
32simprbi 499 . . 3 ( :(𝑀 × 𝑀)⟶𝑀 → ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀)
41, 3syl 17 . 2 ( ∈ ( clIntOp ‘𝑀) → ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀)
5 elfvex 6696 . . 3 ( ∈ ( clIntOp ‘𝑀) → 𝑀 ∈ V)
6 iscllaw 44081 . . 3 (( ∈ ( clIntOp ‘𝑀) ∧ 𝑀 ∈ V) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
75, 6mpdan 685 . 2 ( ∈ ( clIntOp ‘𝑀) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
84, 7mpbird 259 1 ( ∈ ( clIntOp ‘𝑀) → clLaw 𝑀)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2107  ∀wral 3136  Vcvv 3493   class class class wbr 5057   × cxp 5546   Fn wfn 6343  ⟶wf 6344  ‘cfv 6348  (class class class)co 7148   clLaw ccllaw 44075   clIntOp cclintop 44089 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-map 8400  df-cllaw 44078  df-intop 44091  df-clintop 44092 This theorem is referenced by:  assintopcllaw  44104
 Copyright terms: Public domain W3C validator