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Theorem copissgrp 45996
Description: A structure with a constant group addition operation is a semigroup if the constant is contained in the base set. (Contributed by AV, 16-Feb-2020.)
Hypotheses
Ref Expression
copissgrp.b 𝐵 = (Base‘𝑀)
copissgrp.p (+g𝑀) = (𝑥𝐵, 𝑦𝐵𝐶)
copissgrp.n (𝜑𝐵 ≠ ∅)
copissgrp.c (𝜑𝐶𝐵)
Assertion
Ref Expression
copissgrp (𝜑𝑀 ∈ Smgrp)
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝑀   𝜑,𝑥,𝑦
Allowed substitution hint:   𝑀(𝑦)

Proof of Theorem copissgrp
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 copissgrp.b . . 3 𝐵 = (Base‘𝑀)
2 copissgrp.p . . 3 (+g𝑀) = (𝑥𝐵, 𝑦𝐵𝐶)
3 copissgrp.n . . 3 (𝜑𝐵 ≠ ∅)
4 copissgrp.c . . . 4 (𝜑𝐶𝐵)
54adantr 481 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐶𝐵)
61, 2, 3, 5opmpoismgm 45995 . 2 (𝜑𝑀 ∈ Mgm)
7 eqidd 2738 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑥𝐵, 𝑦𝐵𝐶) = (𝑥𝐵, 𝑦𝐵𝐶))
8 eqidd 2738 . . . . . . 7 (((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝐶𝑦 = 𝑐)) → 𝐶 = 𝐶)
9 simpl 483 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝐶𝐵)
10 simpr3 1196 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑐𝐵)
117, 8, 9, 10, 9ovmpod 7501 . . . . . 6 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = 𝐶)
12 eqidd 2738 . . . . . . 7 (((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝑎𝑦 = 𝐶)) → 𝐶 = 𝐶)
13 simpr1 1194 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑎𝐵)
147, 12, 13, 9, 9ovmpod 7501 . . . . . 6 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶) = 𝐶)
1511, 14eqtr4d 2780 . . . . 5 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶))
164, 15sylan 580 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶))
17 eqidd 2738 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑥𝐵, 𝑦𝐵𝐶) = (𝑥𝐵, 𝑦𝐵𝐶))
18 eqidd 2738 . . . . . 6 (((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → 𝐶 = 𝐶)
19 simpr1 1194 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑎𝐵)
20 simpr2 1195 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑏𝐵)
214adantr 481 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝐶𝐵)
2217, 18, 19, 20, 21ovmpod 7501 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏) = 𝐶)
2322oveq1d 7366 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐))
24 eqidd 2738 . . . . . 6 (((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝑏𝑦 = 𝑐)) → 𝐶 = 𝐶)
25 simpr3 1196 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑐𝐵)
2617, 24, 20, 25, 21ovmpod 7501 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = 𝐶)
2726oveq2d 7367 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐)) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶))
2816, 23, 273eqtr4d 2787 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐)))
2928ralrimivvva 3198 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵𝑐𝐵 ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐)))
302eqcomi 2746 . . 3 (𝑥𝐵, 𝑦𝐵𝐶) = (+g𝑀)
311, 30issgrp 18506 . 2 (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐))))
326, 29, 31sylanbrc 583 1 (𝜑𝑀 ∈ Smgrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2941  wral 3062  c0 4280  cfv 6493  (class class class)co 7351  cmpo 7353  Basecbs 17042  +gcplusg 17092  Mgmcmgm 18454  Smgrpcsgrp 18504
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-mgm 18456  df-sgrp 18505
This theorem is referenced by:  cznrng  46147
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