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Theorem opifismgmdc 12808
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 3582 . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → if(𝜓, 𝐶, 𝐷) ∈ 𝐵)
54ralrimivva 2569 . . . . 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 6221 . . . 4 ((∀𝑥𝐵𝑦𝐵 if(𝜓, 𝐶, 𝐷) ∈ 𝐵𝑎𝐵𝑏𝐵) → (𝑎(+g𝑀)𝑏) ∈ 𝐵)
116, 7, 8, 10syl3anc 1248 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵)) → (𝑎(+g𝑀)𝑏) ∈ 𝐵)
1211ralrimivva 2569 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵)
13 opifismgm.m . . 3 (𝜑 → ∃𝑥 𝑥𝐵)
14 opifismgm.b . . . . 5 𝐵 = (Base‘𝑀)
15 eqid 2187 . . . . 5 (+g𝑀) = (+g𝑀)
1614, 15ismgmn0 12795 . . . 4 (𝑥𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑎𝐵𝑏𝐵 (𝑎(+g𝑀)𝑏) ∈ 𝐵))
1716exlimiv 1608 . . 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 835   = wceq 1363  wex 1502  wcel 2158  wral 2465  ifcif 3546  cfv 5228  (class class class)co 5888  cmpo 5890  Basecbs 12475  +gcplusg 12550  Mgmcmgm 12791
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 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-inn 8933  df-2 8991  df-ndx 12478  df-slot 12479  df-base 12481  df-plusg 12563  df-mgm 12793
This theorem is referenced by: (None)
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