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Theorem opifismgm 17861
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 4470 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
43ralrimivva 3156 . . . . 5 (𝜑 → ∀𝑥𝐵𝑦𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
54adantr 484 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ∀𝑥𝐵𝑦𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
6 simprl 770 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
7 simprr 772 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
8 opifismgm.p . . . . 5 (+g𝑀) = (𝑥𝐵, 𝑦𝐵 ↦ if(𝜓, 𝐶, 𝐷))
98ovmpoelrn 7752 . . . 4 ((∀𝑥𝐵𝑦𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵𝑎𝐵𝑏𝐵) → (𝑎(+g𝑀)𝑏) ∈ 𝐵)
105, 6, 7, 9syl3anc 1368 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑀)𝑏) ∈ 𝐵)
1110ralrimivva 3156 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵)
12 opifismgm.n . . 3 (𝜑𝐵 ≠ ∅)
13 n0 4260 . . . 4 (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥𝐵)
14 opifismgm.b . . . . . 6 𝐵 = (Base‘𝑀)
15 eqid 2798 . . . . . 6 (+g𝑀) = (+g𝑀)
1614, 15ismgmn0 17846 . . . . 5 (𝑥𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))
1716exlimiv 1931 . . . 4 (∃𝑥 𝑥𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))
1813, 17sylbi 220 . . 3 (𝐵 ≠ ∅ → (𝑀 ∈ Mgm ↔ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))
1912, 18syl 17 . 2 (𝜑 → (𝑀 ∈ Mgm ↔ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))
2011, 19mpbird 260 1 (𝜑𝑀 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2111  wne 2987  wral 3106  c0 4243  ifcif 4425  cfv 6324  (class class class)co 7135  cmpo 7137  Basecbs 16475  +gcplusg 16557  Mgmcmgm 17842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-mgm 17844
This theorem is referenced by:  mgm2nsgrplem1  18075  sgrp2nmndlem1  18080
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