Theorem ghmcmn 19779

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 19745	. . . 4 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
9	7, 8	syl 17	. . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd)
10	1, 2, 3, 4, 5, 6, 9	mhmmnd 19013	. 2 ⊢ (𝜑 → 𝐻 ∈ Mnd)
11		simp-6l 786	. . . . . . . . . . 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 783	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → 𝑎 ∈ 𝑋)
14		simplr 768	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → 𝑏 ∈ 𝑋)
15	2, 4	cmncom 19746	. . . . . . . . . 10 ⊢ ((𝐺 ∈ CMnd ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
16	12, 13, 14, 15	syl3anc 1369	. . . . . . . . 9 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → (𝑎 + 𝑏) = (𝑏 + 𝑎))
17	16	fveq2d 6895	. . . . . . . 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 1161	. . . . . . . . 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 19011	. . . . . . . 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 19011	. . . . . . . 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 2776	. . . . . . 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 775	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → (𝐹‘𝑎) = 𝑖)
23		simpr 484	. . . . . . . 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 2776	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) ∧ 𝑏 ∈ 𝑋) ∧ (𝐹‘𝑏) = 𝑗) → (𝑖 ⨣ 𝑗) = (𝑗 ⨣ 𝑖))
27		foelcdmi 6954	. . . . . . . 8 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑗 ∈ 𝑌) → ∃𝑏 ∈ 𝑋 (𝐹‘𝑏) = 𝑗)
28	6, 27	sylan 579	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ∃𝑏 ∈ 𝑋 (𝐹‘𝑏) = 𝑗)
29	28	ad5ant13 756	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) → ∃𝑏 ∈ 𝑋 (𝐹‘𝑏) = 𝑗)
30	26, 29	r19.29a 3158	. . . . 5 ⊢ (((((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) ∧ 𝑎 ∈ 𝑋) ∧ (𝐹‘𝑎) = 𝑖) → (𝑖 ⨣ 𝑗) = (𝑗 ⨣ 𝑖))
31		foelcdmi 6954	. . . . . . 7 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑖 ∈ 𝑌) → ∃𝑎 ∈ 𝑋 (𝐹‘𝑎) = 𝑖)
32	6, 31	sylan 579	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → ∃𝑎 ∈ 𝑋 (𝐹‘𝑎) = 𝑖)
33	32	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) → ∃𝑎 ∈ 𝑋 (𝐹‘𝑎) = 𝑖)
34	30, 33	r19.29a 3158	. . . 4 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑌) ∧ 𝑗 ∈ 𝑌) → (𝑖 ⨣ 𝑗) = (𝑗 ⨣ 𝑖))
35	34	anasss 466	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝑌 ∧ 𝑗 ∈ 𝑌)) → (𝑖 ⨣ 𝑗) = (𝑗 ⨣ 𝑖))
36	35	ralrimivva 3196	. 2 ⊢ (𝜑 → ∀𝑖 ∈ 𝑌 ∀𝑗 ∈ 𝑌 (𝑖 ⨣ 𝑗) = (𝑗 ⨣ 𝑖))
37	3, 5	iscmn 19737	. 2 ⊢ (𝐻 ∈ CMnd ↔ (𝐻 ∈ Mnd ∧ ∀𝑖 ∈ 𝑌 ∀𝑗 ∈ 𝑌 (𝑖 ⨣ 𝑗) = (𝑗 ⨣ 𝑖)))
38	10, 36, 37	sylanbrc 582	1 ⊢ (𝜑 → 𝐻 ∈ CMnd)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∀wral 3057  ∃wrex 3066  –onto→wfo 6540  ‘cfv 6542  (class class class)co 7414  Basecbs 17173  +_gcplusg 17226  Mndcmnd 18687  CMndccmn 19728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-riota 7370  df-ov 7417  df-0g 17416  df-mgm 18593  df-sgrp 18672  df-mnd 18688  df-cmn 19730

This theorem is referenced by:  ghmabl  19780