Theorem copissgrp 48815

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 485	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐵)
6	1, 2, 3, 5	opmpoismgm 48814	. 2 ⊢ (𝜑 → 𝑀 ∈ Mgm)
7		eqidd 2770	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐵 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶))
8		eqidd 2770	. . . . . . 7 ⊢ (((𝐶 ∈ 𝐵 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ (𝑥 = 𝐶 ∧ 𝑦 = 𝑐)) → 𝐶 = 𝐶)
9		simpl 487	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐵 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝐶 ∈ 𝐵)
10		simpr3 1213	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐵 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝑐 ∈ 𝐵)
11	7, 8, 9, 10, 9	ovmpod 7560	. . . . . 6 ⊢ ((𝐶 ∈ 𝐵 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝐶(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = 𝐶)
12		eqidd 2770	. . . . . . 7 ⊢ (((𝐶 ∈ 𝐵 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝐶)) → 𝐶 = 𝐶)
13		simpr1 1211	. . . . . . 7 ⊢ ((𝐶 ∈ 𝐵 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝑎 ∈ 𝐵)
14	7, 12, 13, 9, 9	ovmpod 7560	. . . . . 6 ⊢ ((𝐶 ∈ 𝐵 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝐶) = 𝐶)
15	11, 14	eqtr4d 2807	. . . . 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 591	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝐶(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝐶))
17		eqidd 2770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶))
18		eqidd 2770	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑏)) → 𝐶 = 𝐶)
19		simpr1 1211	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝑎 ∈ 𝐵)
20		simpr2 1212	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
21	4	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝐶 ∈ 𝐵)
22	17, 18, 19, 20, 21	ovmpod 7560	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑎(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑏) = 𝐶)
23	22	oveq1d 7423	. . . 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 2770	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) ∧ (𝑥 = 𝑏 ∧ 𝑦 = 𝑐)) → 𝐶 = 𝐶)
25		simpr3 1213	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → 𝑐 ∈ 𝐵)
26	17, 24, 20, 25, 21	ovmpod 7560	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → (𝑏(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)𝑐) = 𝐶)
27	26	oveq2d 7424	. . . 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 2814	. . 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 3217	. 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 2778	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶) = (+g‘𝑀)
31	1, 30	issgrp 18774	. 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 594	1 ⊢ (𝜑 → 𝑀 ∈ Smgrp)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∅c0 4294  ‘cfv 6533  (class class class)co 7408   ∈ cmpo 7410  Basecbs 17265  +_gcplusg 17306  Mgmcmgm 18692  Smgrpcsgrp 18772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-mgm 18694  df-sgrp 18773

This theorem is referenced by:  cznrng  48908