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Theorem opifismgm 17466
Description: A structure with a group addition operation expressed by a conditional operator is a magma if both values of the conditional operator are contained in the base set. (Contributed by AV, 9-Feb-2020.)
Hypotheses
Ref Expression
opifismgm.b 𝐵 = (Base‘𝑀)
opifismgm.p (+g𝑀) = (𝑥𝐵, 𝑦𝐵 ↦ if(𝜓, 𝐶, 𝐷))
opifismgm.n (𝜑𝐵 ≠ ∅)
opifismgm.c ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐶𝐵)
opifismgm.d ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐷𝐵)
Assertion
Ref Expression
opifismgm (𝜑𝑀 ∈ Mgm)
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑀   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑀(𝑦)

Proof of Theorem opifismgm
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opifismgm.c . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐶𝐵)
2 opifismgm.d . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐷𝐵)
31, 2ifcld 4271 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
43ralrimivva 3120 . . . . 5 (𝜑 → ∀𝑥𝐵𝑦𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
54adantr 466 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ∀𝑥𝐵𝑦𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
6 simprl 754 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
7 simprr 756 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
8 opifismgm.p . . . . 5 (+g𝑀) = (𝑥𝐵, 𝑦𝐵 ↦ if(𝜓, 𝐶, 𝐷))
98ovmpt2elrn 7395 . . . 4 ((∀𝑥𝐵𝑦𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵𝑎𝐵𝑏𝐵) → (𝑎(+g𝑀)𝑏) ∈ 𝐵)
105, 6, 7, 9syl3anc 1476 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑀)𝑏) ∈ 𝐵)
1110ralrimivva 3120 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵)
12 opifismgm.n . . 3 (𝜑𝐵 ≠ ∅)
13 n0 4079 . . . 4 (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥𝐵)
14 opifismgm.b . . . . . 6 𝐵 = (Base‘𝑀)
15 eqid 2771 . . . . . 6 (+g𝑀) = (+g𝑀)
1614, 15ismgmn0 17452 . . . . 5 (𝑥𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))
1716exlimiv 2010 . . . 4 (∃𝑥 𝑥𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))
1813, 17sylbi 207 . . 3 (𝐵 ≠ ∅ → (𝑀 ∈ Mgm ↔ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))
1912, 18syl 17 . 2 (𝜑 → (𝑀 ∈ Mgm ↔ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))
2011, 19mpbird 247 1 (𝜑𝑀 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  wne 2943  wral 3061  c0 4063  ifcif 4226  cfv 6030  (class class class)co 6796  cmpt2 6798  Basecbs 16064  +gcplusg 16149  Mgmcmgm 17448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-mgm 17450
This theorem is referenced by:  mgm2nsgrplem1  17613  sgrp2nmndlem1  17618
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