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Theorem copissgrp 44823
 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 484 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐶𝐵)
61, 2, 3, 5opmpoismgm 44822 . 2 (𝜑𝑀 ∈ Mgm)
7 eqidd 2759 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑥𝐵, 𝑦𝐵𝐶) = (𝑥𝐵, 𝑦𝐵𝐶))
8 eqidd 2759 . . . . . . 7 (((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝐶𝑦 = 𝑐)) → 𝐶 = 𝐶)
9 simpl 486 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝐶𝐵)
10 simpr3 1193 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑐𝐵)
117, 8, 9, 10, 9ovmpod 7302 . . . . . 6 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = 𝐶)
12 eqidd 2759 . . . . . . 7 (((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝑎𝑦 = 𝐶)) → 𝐶 = 𝐶)
13 simpr1 1191 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑎𝐵)
147, 12, 13, 9, 9ovmpod 7302 . . . . . 6 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶) = 𝐶)
1511, 14eqtr4d 2796 . . . . 5 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶))
164, 15sylan 583 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶))
17 eqidd 2759 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑥𝐵, 𝑦𝐵𝐶) = (𝑥𝐵, 𝑦𝐵𝐶))
18 eqidd 2759 . . . . . 6 (((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → 𝐶 = 𝐶)
19 simpr1 1191 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑎𝐵)
20 simpr2 1192 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑏𝐵)
214adantr 484 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝐶𝐵)
2217, 18, 19, 20, 21ovmpod 7302 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏) = 𝐶)
2322oveq1d 7170 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐))
24 eqidd 2759 . . . . . 6 (((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝑏𝑦 = 𝑐)) → 𝐶 = 𝐶)
25 simpr3 1193 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑐𝐵)
2617, 24, 20, 25, 21ovmpod 7302 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = 𝐶)
2726oveq2d 7171 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐)) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶))
2816, 23, 273eqtr4d 2803 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐)))
2928ralrimivvva 3121 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵𝑐𝐵 ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐)))
302eqcomi 2767 . . 3 (𝑥𝐵, 𝑦𝐵𝐶) = (+g𝑀)
311, 30issgrp 17973 . 2 (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐))))
326, 29, 31sylanbrc 586 1 (𝜑𝑀 ∈ Smgrp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∅c0 4227  ‘cfv 6339  (class class class)co 7155   ∈ cmpo 7157  Basecbs 16546  +gcplusg 16628  Mgmcmgm 17921  Smgrpcsgrp 17971 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7698  df-2nd 7699  df-mgm 17923  df-sgrp 17972 This theorem is referenced by:  cznrng  44974
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