Theorem mhmid 18019

Description: A surjective monoid morphism preserves identity element. (Contributed by Thierry Arnoux, 25-Jan-2020.)

Hypotheses

Ref	Expression
ghmgrp.f	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
ghmgrp.x	⊢ 𝑋 = (Base‘𝐺)
ghmgrp.y	⊢ 𝑌 = (Base‘𝐻)
ghmgrp.p	⊢ + = (+g‘𝐺)
ghmgrp.q	⊢ ⨣ = (+g‘𝐻)
ghmgrp.1	⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
mhmmnd.3	⊢ (𝜑 → 𝐺 ∈ Mnd)
mhmid.0	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
mhmid	⊢ (𝜑 → (𝐹‘ 0 ) = (0g‘𝐻))

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥, + ,𝑦   𝑥,𝐻,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥, ⨣ ,𝑦   𝜑,𝑥,𝑦   𝑥, 0 ,𝑦

Proof of Theorem mhmid

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

Step	Hyp	Ref	Expression
1		ghmgrp.y	. 2 ⊢ 𝑌 = (Base‘𝐻)
2		eqid 2771	. 2 ⊢ (0g‘𝐻) = (0g‘𝐻)
3		ghmgrp.q	. 2 ⊢ ⨣ = (+g‘𝐻)
4		ghmgrp.1	. . . 4 ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
5		fof 6416	. . . 4 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐹:𝑋⟶𝑌)
7		mhmmnd.3	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd)
8		ghmgrp.x	. . . . 5 ⊢ 𝑋 = (Base‘𝐺)
9		mhmid.0	. . . . 5 ⊢ 0 = (0g‘𝐺)
10	8, 9	mndidcl 17788	. . . 4 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝑋)
11	7, 10	syl 17	. . 3 ⊢ (𝜑 → 0 ∈ 𝑋)
12	6, 11	ffvelrnd 6675	. 2 ⊢ (𝜑 → (𝐹‘ 0 ) ∈ 𝑌)
13		simplll 763	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝜑)
14		ghmgrp.f	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
15	13, 14	syl3an1 1144	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
16	7	ad3antrrr 718	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝐺 ∈ Mnd)
17	16, 10	syl 17	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 0 ∈ 𝑋)
18		simplr 757	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝑖 ∈ 𝑋)
19	15, 17, 18	mhmlem 18018	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘( 0 + 𝑖)) = ((𝐹‘ 0 ) ⨣ (𝐹‘𝑖)))
20		ghmgrp.p	. . . . . . . 8 ⊢ + = (+g‘𝐺)
21	8, 20, 9	mndlid 17791	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝑖 ∈ 𝑋) → ( 0 + 𝑖) = 𝑖)
22	16, 18, 21	syl2anc 576	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ( 0 + 𝑖) = 𝑖)
23	22	fveq2d 6500	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘( 0 + 𝑖)) = (𝐹‘𝑖))
24	19, 23	eqtr3d 2809	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘ 0 ) ⨣ (𝐹‘𝑖)) = (𝐹‘𝑖))
25		simpr 477	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘𝑖) = 𝑎)
26	25	oveq2d 6990	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘ 0 ) ⨣ (𝐹‘𝑖)) = ((𝐹‘ 0 ) ⨣ 𝑎))
27	24, 26, 25	3eqtr3d 2815	. . 3 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘ 0 ) ⨣ 𝑎) = 𝑎)
28		foelrni 6554	. . . 4 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
29	4, 28	sylan 572	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
30	27, 29	r19.29a 3227	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → ((𝐹‘ 0 ) ⨣ 𝑎) = 𝑎)
31	15, 18, 17	mhmlem 18018	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(𝑖 + 0 )) = ((𝐹‘𝑖) ⨣ (𝐹‘ 0 )))
32	8, 20, 9	mndrid 17792	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝑖 ∈ 𝑋) → (𝑖 + 0 ) = 𝑖)
33	16, 18, 32	syl2anc 576	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝑖 + 0 ) = 𝑖)
34	33	fveq2d 6500	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(𝑖 + 0 )) = (𝐹‘𝑖))
35	31, 34	eqtr3d 2809	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘𝑖) ⨣ (𝐹‘ 0 )) = (𝐹‘𝑖))
36	25	oveq1d 6989	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘𝑖) ⨣ (𝐹‘ 0 )) = (𝑎 ⨣ (𝐹‘ 0 )))
37	35, 36, 25	3eqtr3d 2815	. . 3 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝑎 ⨣ (𝐹‘ 0 )) = 𝑎)
38	37, 29	r19.29a 3227	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → (𝑎 ⨣ (𝐹‘ 0 )) = 𝑎)
39	1, 2, 3, 12, 30, 38	ismgmid2 17747	1 ⊢ (𝜑 → (𝐹‘ 0 ) = (0g‘𝐻))

Syntax hints:   → wi 4   ∧ wa 387   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051  ∃wrex 3082  ⟶wf 6181  –onto→wfo 6183  ‘cfv 6185  (class class class)co 6974  Basecbs 16337  +_gcplusg 16419  0_gc0g 16567  Mndcmnd 17774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rmo 3089  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-fo 6191  df-fv 6193  df-riota 6935  df-ov 6977  df-0g 16569  df-mgm 17722  df-sgrp 17764  df-mnd 17775

This theorem is referenced by:  mhmfmhm  18021  ghmgrp  18022