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Theorem opifismgm 18592
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 4569 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
43ralrimivva 3194 . . . . 5 (𝜑 → ∀𝑥𝐵𝑦𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
54adantr 480 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ∀𝑥𝐵𝑦𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
6 simprl 768 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
7 simprr 770 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
8 opifismgm.p . . . . 5 (+g𝑀) = (𝑥𝐵, 𝑦𝐵 ↦ if(𝜓, 𝐶, 𝐷))
98ovmpoelrn 8057 . . . 4 ((∀𝑥𝐵𝑦𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵𝑎𝐵𝑏𝐵) → (𝑎(+g𝑀)𝑏) ∈ 𝐵)
105, 6, 7, 9syl3anc 1368 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑀)𝑏) ∈ 𝐵)
1110ralrimivva 3194 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵)
12 opifismgm.n . . 3 (𝜑𝐵 ≠ ∅)
13 n0 4341 . . . 4 (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥𝐵)
14 opifismgm.b . . . . . 6 𝐵 = (Base‘𝑀)
15 eqid 2726 . . . . . 6 (+g𝑀) = (+g𝑀)
1614, 15ismgmn0 18575 . . . . 5 (𝑥𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))
1716exlimiv 1925 . . . 4 (∃𝑥 𝑥𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))
1813, 17sylbi 216 . . 3 (𝐵 ≠ ∅ → (𝑀 ∈ Mgm ↔ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))
1912, 18syl 17 . 2 (𝜑 → (𝑀 ∈ Mgm ↔ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))
2011, 19mpbird 257 1 (𝜑𝑀 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wex 1773  wcel 2098  wne 2934  wral 3055  c0 4317  ifcif 4523  cfv 6537  (class class class)co 7405  cmpo 7407  Basecbs 17153  +gcplusg 17206  Mgmcmgm 18571
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-mgm 18573
This theorem is referenced by:  mgm2nsgrplem1  18843  sgrp2nmndlem1  18848
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