Theorem ghmcmn 19610

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 19579	. . . 4 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
9	7, 8	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd)
10	1, 2, 3, 4, 5, 6, 9	mhmmnd 18869	. 2 ⊢ (𝜑 → 𝐻 ∈ Mnd)
11		simp-6l 785	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → 𝜑)
12	11, 7	syl 17	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → 𝐺 ∈ CMnd)
13		simp-4r 782	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → 𝑎 ∈ 𝑋)
14		simplr 767	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → 𝑏 ∈ 𝑋)
15	2, 4	cmncom 19580	. . . . . . . . . 10 ⊢ ((𝐺 ∈ CMnd ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
16	12, 13, 14, 15	syl3anc 1371	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
17	16	fveq2d 6846	. . . . . . . 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 1163	. . . . . . . . 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 18867	. . . . . . . 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 18867	. . . . . . . 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 2784	. . . . . . 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 774	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → (𝐹‘𝑎) = 𝑖)
23		simpr 485	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → (𝐹‘𝑏) = 𝑗)
24	22, 23	oveq12d 7375	. . . . . . 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 7375	. . . . . . 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 2784	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → (𝑖 ⨣ 𝑗) = (𝑗 ⨣ 𝑖))
27		foelcdmi 6904	. . . . . . . 8 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑗 ∈ 𝑌) → ∃𝑏 ∈ 𝑋 (𝐹‘𝑏) = 𝑗)
28	6, 27	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ∃𝑏 ∈ 𝑋 (𝐹‘𝑏) = 𝑗)
29	28	ad5ant13 755	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) → ∃𝑏 ∈ 𝑋 (𝐹‘𝑏) = 𝑗)
30	26, 29	r19.29a 3159	. . . . 5 ⊢ (((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) → (𝑖 ⨣ 𝑗) = (𝑗 ⨣ 𝑖))
31		foelcdmi 6904	. . . . . . 7 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑖 ∈ 𝑌) → ∃𝑎 ∈ 𝑋 (𝐹‘𝑎) = 𝑖)
32	6, 31	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → ∃𝑎 ∈ 𝑋 (𝐹‘𝑎) = 𝑖)
33	32	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) → ∃𝑎 ∈ 𝑋 (𝐹‘𝑎) = 𝑖)
34	30, 33	r19.29a 3159	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) → (𝑖 ⨣ 𝑗) = (𝑗 ⨣ 𝑖))
35	34	anasss 467	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑌 ∧ 𝑗 ∈ 𝑌)) → (𝑖 ⨣ 𝑗) = (𝑗 ⨣ 𝑖))
36	35	ralrimivva 3197	. 2 ⊢ (𝜑 → ∀𝑖 ∈ 𝑌 ∀𝑗 ∈ 𝑌 (𝑖 ⨣ 𝑗) = (𝑗 ⨣ 𝑖))
37	3, 5	iscmn 19571	. 2 ⊢ (𝐻 ∈ CMnd ↔ (𝐻 ∈ Mnd ∧ ∀𝑖 ∈ 𝑌 ∀𝑗 ∈ 𝑌 (𝑖 ⨣ 𝑗) = (𝑗 ⨣ 𝑖)))
38	10, 36, 37	sylanbrc 583	1 ⊢ (𝜑 → 𝐻 ∈ CMnd)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  –onto→wfo 6494  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  +_gcplusg 17133  Mndcmnd 18556  CMndccmn 19562

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504  df-riota 7313  df-ov 7360  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-cmn 19564

This theorem is referenced by:  ghmabl  19611