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Theorem copissgrp 43441
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 ∈ SGrp)
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 473 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐶𝐵)
61, 2, 3, 5opmpoismgm 43440 . 2 (𝜑𝑀 ∈ Mgm)
7 eqidd 2780 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑥𝐵, 𝑦𝐵𝐶) = (𝑥𝐵, 𝑦𝐵𝐶))
8 eqidd 2780 . . . . . . 7 (((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝐶𝑦 = 𝑐)) → 𝐶 = 𝐶)
9 simpl 475 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝐶𝐵)
10 simpr3 1176 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑐𝐵)
117, 8, 9, 10, 9ovmpod 7118 . . . . . 6 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = 𝐶)
12 eqidd 2780 . . . . . . 7 (((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝑎𝑦 = 𝐶)) → 𝐶 = 𝐶)
13 simpr1 1174 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑎𝐵)
147, 12, 13, 9, 9ovmpod 7118 . . . . . 6 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶) = 𝐶)
1511, 14eqtr4d 2818 . . . . 5 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶))
164, 15sylan 572 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶))
17 eqidd 2780 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑥𝐵, 𝑦𝐵𝐶) = (𝑥𝐵, 𝑦𝐵𝐶))
18 eqidd 2780 . . . . . 6 (((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → 𝐶 = 𝐶)
19 simpr1 1174 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑎𝐵)
20 simpr2 1175 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑏𝐵)
214adantr 473 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝐶𝐵)
2217, 18, 19, 20, 21ovmpod 7118 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏) = 𝐶)
2322oveq1d 6991 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐))
24 eqidd 2780 . . . . . 6 (((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝑏𝑦 = 𝑐)) → 𝐶 = 𝐶)
25 simpr3 1176 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑐𝐵)
2617, 24, 20, 25, 21ovmpod 7118 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = 𝐶)
2726oveq2d 6992 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐)) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶))
2816, 23, 273eqtr4d 2825 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐)))
2928ralrimivvva 3143 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵𝑐𝐵 ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐)))
302eqcomi 2788 . . 3 (𝑥𝐵, 𝑦𝐵𝐶) = (+g𝑀)
311, 30issgrp 17753 . 2 (𝑀 ∈ SGrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐))))
326, 29, 31sylanbrc 575 1 (𝜑𝑀 ∈ SGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wcel 2050  wne 2968  wral 3089  c0 4179  cfv 6188  (class class class)co 6976  cmpo 6978  Basecbs 16339  +gcplusg 16421  Mgmcmgm 17708  SGrpcsgrp 17751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-1st 7501  df-2nd 7502  df-mgm 17710  df-sgrp 17752
This theorem is referenced by:  cznrng  43588
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