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Theorem ismgm 17012
Description: The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
Hypotheses
Ref Expression
ismgm.b 𝐵 = (Base‘𝑀)
ismgm.o = (+g𝑀)
Assertion
Ref Expression
ismgm (𝑀𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑀,𝑦   𝑥, ,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem ismgm
Dummy variables 𝑏 𝑚 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6098 . . . 4 (Base‘𝑚) ∈ V
21a1i 11 . . 3 (𝑚 = 𝑀 → (Base‘𝑚) ∈ V)
3 fveq2 6088 . . . 4 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
4 ismgm.b . . . 4 𝐵 = (Base‘𝑀)
53, 4syl6eqr 2661 . . 3 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
6 fvex 6098 . . . . 5 (+g𝑚) ∈ V
76a1i 11 . . . 4 ((𝑚 = 𝑀𝑏 = 𝐵) → (+g𝑚) ∈ V)
8 fveq2 6088 . . . . . 6 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
98adantr 479 . . . . 5 ((𝑚 = 𝑀𝑏 = 𝐵) → (+g𝑚) = (+g𝑀))
10 ismgm.o . . . . 5 = (+g𝑀)
119, 10syl6eqr 2661 . . . 4 ((𝑚 = 𝑀𝑏 = 𝐵) → (+g𝑚) = )
12 simplr 787 . . . . 5 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → 𝑏 = 𝐵)
13 oveq 6533 . . . . . . . 8 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
1413adantl 480 . . . . . . 7 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (𝑥𝑜𝑦) = (𝑥 𝑦))
1514, 12eleq12d 2681 . . . . . 6 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → ((𝑥𝑜𝑦) ∈ 𝑏 ↔ (𝑥 𝑦) ∈ 𝐵))
1612, 15raleqbidv 3128 . . . . 5 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
1712, 16raleqbidv 3128 . . . 4 (((𝑚 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
187, 11, 17sbcied2 3439 . . 3 ((𝑚 = 𝑀𝑏 = 𝐵) → ([(+g𝑚) / 𝑜]𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
192, 5, 18sbcied2 3439 . 2 (𝑚 = 𝑀 → ([(Base‘𝑚) / 𝑏][(+g𝑚) / 𝑜]𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
20 df-mgm 17011 . 2 Mgm = {𝑚[(Base‘𝑚) / 𝑏][(+g𝑚) / 𝑜]𝑥𝑏𝑦𝑏 (𝑥𝑜𝑦) ∈ 𝑏}
2119, 20elab2g 3321 1 (𝑀𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  Vcvv 3172  [wsbc 3401  cfv 5790  (class class class)co 6527  Basecbs 15641  +gcplusg 15714  Mgmcmgm 17009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-nul 4712
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798  df-ov 6530  df-mgm 17011
This theorem is referenced by:  ismgmn0  17013  mgmcl  17014  issstrmgm  17021  mgm0  17024  issgrpv  17055  0mgm  41559  ismgmd  41561  mgm2mgm  41648  lidlmmgm  41710
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