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Theorem copissgrp 41579
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 479 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐶𝐵)
61, 2, 3, 5opmpt2ismgm 41578 . 2 (𝜑𝑀 ∈ Mgm)
7 eqidd 2610 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑥𝐵, 𝑦𝐵𝐶) = (𝑥𝐵, 𝑦𝐵𝐶))
8 eqidd 2610 . . . . . . 7 (((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝐶𝑦 = 𝑐)) → 𝐶 = 𝐶)
9 simpl 471 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝐶𝐵)
10 simpr3 1061 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑐𝐵)
117, 8, 9, 10, 9ovmpt2d 6663 . . . . . 6 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = 𝐶)
12 eqidd 2610 . . . . . . 7 (((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝑎𝑦 = 𝐶)) → 𝐶 = 𝐶)
13 simpr1 1059 . . . . . . 7 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑎𝐵)
147, 12, 13, 9, 9ovmpt2d 6663 . . . . . 6 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶) = 𝐶)
1511, 14eqtr4d 2646 . . . . 5 ((𝐶𝐵 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶))
164, 15sylan 486 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶))
17 eqidd 2610 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑥𝐵, 𝑦𝐵𝐶) = (𝑥𝐵, 𝑦𝐵𝐶))
18 eqidd 2610 . . . . . 6 (((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → 𝐶 = 𝐶)
19 simpr1 1059 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑎𝐵)
20 simpr2 1060 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑏𝐵)
214adantr 479 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝐶𝐵)
2217, 18, 19, 20, 21ovmpt2d 6663 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏) = 𝐶)
2322oveq1d 6541 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝐶(𝑥𝐵, 𝑦𝐵𝐶)𝑐))
24 eqidd 2610 . . . . . 6 (((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) ∧ (𝑥 = 𝑏𝑦 = 𝑐)) → 𝐶 = 𝐶)
25 simpr3 1061 . . . . . 6 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → 𝑐𝐵)
2617, 24, 20, 25, 21ovmpt2d 6663 . . . . 5 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = 𝐶)
2726oveq2d 6542 . . . 4 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐)) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝐶))
2816, 23, 273eqtr4d 2653 . . 3 ((𝜑 ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐)))
2928ralrimivvva 2954 . 2 (𝜑 → ∀𝑎𝐵𝑏𝐵𝑐𝐵 ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐)))
302eqcomi 2618 . . 3 (𝑥𝐵, 𝑦𝐵𝐶) = (+g𝑀)
311, 30issgrp 17056 . 2 (𝑀 ∈ SGrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑎𝐵𝑏𝐵𝑐𝐵 ((𝑎(𝑥𝐵, 𝑦𝐵𝐶)𝑏)(𝑥𝐵, 𝑦𝐵𝐶)𝑐) = (𝑎(𝑥𝐵, 𝑦𝐵𝐶)(𝑏(𝑥𝐵, 𝑦𝐵𝐶)𝑐))))
326, 29, 31sylanbrc 694 1 (𝜑𝑀 ∈ SGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  c0 3873  cfv 5789  (class class class)co 6526  cmpt2 6528  Basecbs 15643  +gcplusg 15716  Mgmcmgm 17011  SGrpcsgrp 17054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-1st 7036  df-2nd 7037  df-mgm 17013  df-sgrp 17055
This theorem is referenced by:  cznrng  41728
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