Theorem ghmcmn 19904

Description: The image of a commutative monoid 𝐺 under a group homomorphism 𝐹 is a commutative monoid. (Contributed by Thierry Arnoux, 26-Jan-2020.)

Hypotheses

Ref	Expression
ghmabl.x	⊢ 𝑋 = (Base‘𝐺)
ghmabl.y	⊢ 𝑌 = (Base‘𝐻)
ghmabl.p	⊢ + = (+g‘𝐺)
ghmabl.q	⊢ ⨣ = (+g‘𝐻)
ghmabl.f	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
ghmabl.1	⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
ghmcmn.3	⊢ (𝜑 → 𝐺 ∈ CMnd)

Assertion

Ref	Expression
ghmcmn	⊢ (𝜑 → 𝐻 ∈ CMnd)

Distinct variable groups:   𝑥, + ,𝑦   𝑥, ⨣ ,𝑦   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝜑,𝑥,𝑦

Proof of Theorem ghmcmn

Dummy variables 𝑎 𝑏 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ghmabl.f	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
2		ghmabl.x	. . 3 ⊢ 𝑋 = (Base‘𝐺)
3		ghmabl.y	. . 3 ⊢ 𝑌 = (Base‘𝐻)
4		ghmabl.p	. . 3 ⊢ + = (+g‘𝐺)
5		ghmabl.q	. . 3 ⊢ ⨣ = (+g‘𝐻)
6		ghmabl.1	. . 3 ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
7		ghmcmn.3	. . . 4 ⊢ (𝜑 → 𝐺 ∈ CMnd)
8		cmnmnd 19870	. . . 4 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
9	7, 8	syl 18	. . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd)
10	1, 2, 3, 4, 5, 6, 9	mhmmnd 19133	. 2 ⊢ (𝜑 → 𝐻 ∈ Mnd)
11		simp-6l 798	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → 𝜑)
12	11, 7	syl 18	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → 𝐺 ∈ CMnd)
13		simp-4r 795	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → 𝑎 ∈ 𝑋)
14		simplr 780	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → 𝑏 ∈ 𝑋)
15	2, 4	cmncom 19871	. . . . . . . . . 10 ⊢ ((𝐺 ∈ CMnd ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
16	12, 13, 14, 15	syl3anc 1396	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
17	16	fveq2d 6889	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → (𝐹‘(𝑎 + 𝑏)) = (𝐹‘(𝑏 + 𝑎)))
18	11, 1	syl3an1 1179	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
19	18, 13, 14	mhmlem 19131	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → (𝐹‘(𝑎 + 𝑏)) = ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)))
20	18, 14, 13	mhmlem 19131	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → (𝐹‘(𝑏 + 𝑎)) = ((𝐹‘𝑏) ⨣ (𝐹‘𝑎)))
21	17, 19, 20	3eqtr3d 2813	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)) = ((𝐹‘𝑏) ⨣ (𝐹‘𝑎)))
22		simpllr 787	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → (𝐹‘𝑎) = 𝑖)
23		simpr 489	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → (𝐹‘𝑏) = 𝑗)
24	22, 23	oveq12d 7432	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → ((𝐹‘𝑎) ⨣ (𝐹‘𝑏)) = (𝑖 ⨣ 𝑗))
25	23, 22	oveq12d 7432	. . . . . . 7 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → ((𝐹‘𝑏) ⨣ (𝐹‘𝑎)) = (𝑗 ⨣ 𝑖))
26	21, 24, 25	3eqtr3d 2813	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → (𝑖 ⨣ 𝑗) = (𝑗 ⨣ 𝑖))
27		foelcdmi 6946	. . . . . . . 8 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑗 ∈ 𝑌) → ∃𝑏 ∈ 𝑋 (𝐹‘𝑏) = 𝑗)
28	6, 27	sylan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ∃𝑏 ∈ 𝑋 (𝐹‘𝑏) = 𝑗)
29	28	ad5ant13 768	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) → ∃𝑏 ∈ 𝑋 (𝐹‘𝑏) = 𝑗)
30	26, 29	r19.29a 3180	. . . . 5 ⊢ (((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) → (𝑖 ⨣ 𝑗) = (𝑗 ⨣ 𝑖))
31		foelcdmi 6946	. . . . . . 7 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑖 ∈ 𝑌) → ∃𝑎 ∈ 𝑋 (𝐹‘𝑎) = 𝑖)
32	6, 31	sylan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → ∃𝑎 ∈ 𝑋 (𝐹‘𝑎) = 𝑖)
33	32	adantr 485	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) → ∃𝑎 ∈ 𝑋 (𝐹‘𝑎) = 𝑖)
34	30, 33	r19.29a 3180	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) → (𝑖 ⨣ 𝑗) = (𝑗 ⨣ 𝑖))
35	34	anasss 471	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑌 ∧ 𝑗 ∈ 𝑌)) → (𝑖 ⨣ 𝑗) = (𝑗 ⨣ 𝑖))
36	35	ralrimivva 3215	. 2 ⊢ (𝜑 → ∀𝑖 ∈ 𝑌 ∀𝑗 ∈ 𝑌 (𝑖 ⨣ 𝑗) = (𝑗 ⨣ 𝑖))
37	3, 5	iscmn 19862	. 2 ⊢ (𝐻 ∈ CMnd ↔ (𝐻 ∈ Mnd ∧ ∀𝑖 ∈ 𝑌 ∀𝑗 ∈ 𝑌 (𝑖 ⨣ 𝑗) = (𝑗 ⨣ 𝑖)))
38	10, 36, 37	sylanbrc 594	1 ⊢ (𝜑 → 𝐻 ∈ CMnd)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2150  ∀wral 3086  ∃wrex 3096  –onto→wfo 6538  ‘cfv 6540  (class class class)co 7414  Basecbs 17272  +_gcplusg 17313  Mndcmnd 18795  CMndccmn 19853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-riota 7371  df-ov 7417  df-0g 17497  df-mgm 18701  df-sgrp 18780  df-mnd 18796  df-cmn 19855

This theorem is referenced by:  ghmabl  19905