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Theorem opifismgmdc 12669
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))
opifismgmdc.dc ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → DECID 𝜓)
opifismgm.m (𝜑 → ∃𝑥 𝑥𝐵)
opifismgm.c ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐶𝐵)
opifismgm.d ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐷𝐵)
Assertion
Ref Expression
opifismgmdc (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 opifismgmdc
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)
3 opifismgmdc.dc . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → DECID 𝜓)
41, 2, 3ifcldcd 3569 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
54ralrimivva 2559 . . . . 5 (𝜑 → ∀𝑥𝐵𝑦𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
65adantr 276 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → ∀𝑥𝐵𝑦𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
7 simprl 529 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑎𝐵)
8 simprr 531 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → 𝑏𝐵)
9 opifismgm.p . . . . 5 (+g𝑀) = (𝑥𝐵, 𝑦𝐵 ↦ if(𝜓, 𝐶, 𝐷))
109ovmpoelrn 6201 . . . 4 ((∀𝑥𝐵𝑦𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵𝑎𝐵𝑏𝐵) → (𝑎(+g𝑀)𝑏) ∈ 𝐵)
116, 7, 8, 10syl3anc 1238 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑀)𝑏) ∈ 𝐵)
1211ralrimivva 2559 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵)
13 opifismgm.m . . 3 (𝜑 → ∃𝑥 𝑥𝐵)
14 opifismgm.b . . . . 5 𝐵 = (Base‘𝑀)
15 eqid 2177 . . . . 5 (+g𝑀) = (+g𝑀)
1614, 15ismgmn0 12656 . . . 4 (𝑥𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))
1716exlimiv 1598 . . 3 (∃𝑥 𝑥𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))
1813, 17syl 14 . 2 (𝜑 → (𝑀 ∈ Mgm ↔ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))
1912, 18mpbird 167 1 (𝜑𝑀 ∈ Mgm)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  DECID wdc 834   = wceq 1353  wex 1492  wcel 2148  wral 2455  ifcif 3534  cfv 5211  (class class class)co 5868  cmpo 5870  Basecbs 12432  +gcplusg 12505  Mgmcmgm 12652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-inn 8896  df-2 8954  df-ndx 12435  df-slot 12436  df-base 12438  df-plusg 12518  df-mgm 12654
This theorem is referenced by: (None)
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