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	⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)
copissgrp.n	⊢ (𝜑 → 𝐵 ≠ ∅)
copissgrp.c	⊢ (𝜑 → 𝐶 ∈ 𝐵)

Assertion

Ref	Expression
copissgrp	⊢ (𝜑 → 𝑀 ∈ 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.

Step	Hyp	Ref	Expression
1		copissgrp.b	. . 3 ⊢ 𝐵 = (Base‘𝑀)
2		copissgrp.p	. . 3 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)
3		copissgrp.n	. . 3 ⊢ (𝜑 → 𝐵 ≠ ∅)
4		copissgrp.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
5	4	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐵)
6	1, 2, 3, 5	opmpoismgm 44822	. 2 ⊢ (𝜑 → 𝑀 ∈ Mgm)
7		eqidd 2759	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐵 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶))
8		eqidd 2759	. . . . . . 7 ⊢ (((𝐶 ∈ 𝐵 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ (𝑥 = 𝐶 ∧ 𝑦 = 𝑐)) → 𝐶 = 𝐶)
9		simpl 486	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐵 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝐶 ∈ 𝐵)
10		simpr3 1193	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐵 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝑐 ∈ 𝐵)
11	7, 8, 9, 10, 9	ovmpod 7302	. . . . . 6 ⊢ ((𝐶 ∈ 𝐵 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝐶(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = 𝐶)
12		eqidd 2759	. . . . . . 7 ⊢ (((𝐶 ∈ 𝐵 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝐶)) → 𝐶 = 𝐶)
13		simpr1 1191	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐵 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝑎 ∈ 𝐵)
14	7, 12, 13, 9, 9	ovmpod 7302	. . . . . 6 ⊢ ((𝐶 ∈ 𝐵 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝐶) = 𝐶)
15	11, 14	eqtr4d 2796	. . . . 5 ⊢ ((𝐶 ∈ 𝐵 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝐶(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝐶))
16	4, 15	sylan 583	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝐶(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝐶))
17		eqidd 2759	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶))
18		eqidd 2759	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑏)) → 𝐶 = 𝐶)
19		simpr1 1191	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝑎 ∈ 𝐵)
20		simpr2 1192	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
21	4	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝐶 ∈ 𝐵)
22	17, 18, 19, 20, 21	ovmpod 7302	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏) = 𝐶)
23	22	oveq1d 7170	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏)(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = (𝐶(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐))
24		eqidd 2759	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ (𝑥 = 𝑏 ∧ 𝑦 = 𝑐)) → 𝐶 = 𝐶)
25		simpr3 1193	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝑐 ∈ 𝐵)
26	17, 24, 20, 25, 21	ovmpod 7302	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑏(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = 𝐶)
27	26	oveq2d 7171	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)(𝑏(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)) = (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝐶))
28	16, 23, 27	3eqtr4d 2803	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏)(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)(𝑏(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)))
29	28	ralrimivvva 3121	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏)(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)(𝑏(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐)))
30	2	eqcomi 2767	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (+g‘𝑀)
31	1, 30	issgrp 17973	. 2 ⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ((𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏)(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)(𝑏(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐))))
32	6, 29, 31	sylanbrc 586	1 ⊢ (𝜑 → 𝑀 ∈ Smgrp)

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