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Theorem copissgrp 47118
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 480 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐶𝐵)
61, 2, 3, 5opmpoismgm 47117 . 2 (𝜑𝑀 ∈ Mgm)
7 eqidd 2727 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑥𝐵, 𝑦𝐵𝐶) = (𝑥𝐵, 𝑦𝐵𝐶))
8 eqidd 2727 . . . . . . 7 (((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝐶𝑦 = 𝑐)) → 𝐶 = 𝐶)
9 simpl 482 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝐶𝐵)
10 simpr3 1193 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑐𝐵)
117, 8, 9, 10, 9ovmpod 7556 . . . . . 6 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = 𝐶)
12 eqidd 2727 . . . . . . 7 (((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝑎𝑦 = 𝐶)) → 𝐶 = 𝐶)
13 simpr1 1191 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑎𝐵)
147, 12, 13, 9, 9ovmpod 7556 . . . . . 6 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶) = 𝐶)
1511, 14eqtr4d 2769 . . . . 5 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶))
164, 15sylan 579 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶))
17 eqidd 2727 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑥𝐵, 𝑦𝐵𝐶) = (𝑥𝐵, 𝑦𝐵𝐶))
18 eqidd 2727 . . . . . 6 (((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → 𝐶 = 𝐶)
19 simpr1 1191 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑎𝐵)
20 simpr2 1192 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑏𝐵)
214adantr 480 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝐶𝐵)
2217, 18, 19, 20, 21ovmpod 7556 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏) = 𝐶)
2322oveq1d 7420 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐))
24 eqidd 2727 . . . . . 6 (((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝑏𝑦 = 𝑐)) → 𝐶 = 𝐶)
25 simpr3 1193 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑐𝐵)
2617, 24, 20, 25, 21ovmpod 7556 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = 𝐶)
2726oveq2d 7421 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐)) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶))
2816, 23, 273eqtr4d 2776 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐)))
2928ralrimivvva 3197 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵𝑐𝐵 ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐)))
302eqcomi 2735 . . 3 (𝑥𝐵, 𝑦𝐵𝐶) = (+g𝑀)
311, 30issgrp 18653 . 2 (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐))))
326, 29, 31sylanbrc 582 1 (𝜑𝑀 ∈ Smgrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  wne 2934  wral 3055  c0 4317  cfv 6537  (class class class)co 7405  cmpo 7407  Basecbs 17153  +gcplusg 17206  Mgmcmgm 18571  Smgrpcsgrp 18651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-mgm 18573  df-sgrp 18652
This theorem is referenced by:  cznrng  47211
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