Theorem mhmid 18611

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 2738	. 2 ⊢ (0g‘𝐻) = (0g‘𝐻)
3		ghmgrp.q	. 2 ⊢ ⨣ = (+g‘𝐻)
4		ghmgrp.1	. . . 4 ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
5		fof 6672	. . . 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 18315	. . . 4 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝑋)
11	7, 10	syl 17	. . 3 ⊢ (𝜑 → 0 ∈ 𝑋)
12	6, 11	ffvelrnd 6944	. 2 ⊢ (𝜑 → (𝐹‘ 0 ) ∈ 𝑌)
13		simplll 771	. . . . . . 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 1161	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
16	7	ad3antrrr 726	. . . . . . 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 765	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝑖 ∈ 𝑋)
19	15, 17, 18	mhmlem 18610	. . . . 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 18320	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝑖 ∈ 𝑋) → ( 0 + 𝑖) = 𝑖)
22	16, 18, 21	syl2anc 583	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ( 0 + 𝑖) = 𝑖)
23	22	fveq2d 6760	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘( 0 + 𝑖)) = (𝐹‘𝑖))
24	19, 23	eqtr3d 2780	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘ 0 ) ⨣ (𝐹‘𝑖)) = (𝐹‘𝑖))
25		simpr 484	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘𝑖) = 𝑎)
26	25	oveq2d 7271	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘ 0 ) ⨣ (𝐹‘𝑖)) = ((𝐹‘ 0 ) ⨣ 𝑎))
27	24, 26, 25	3eqtr3d 2786	. . 3 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘ 0 ) ⨣ 𝑎) = 𝑎)
28		foelrni 6813	. . . 4 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
29	4, 28	sylan 579	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
30	27, 29	r19.29a 3217	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → ((𝐹‘ 0 ) ⨣ 𝑎) = 𝑎)
31	15, 18, 17	mhmlem 18610	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(𝑖 + 0 )) = ((𝐹‘𝑖) ⨣ (𝐹‘ 0 )))
32	8, 20, 9	mndrid 18321	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝑖 ∈ 𝑋) → (𝑖 + 0 ) = 𝑖)
33	16, 18, 32	syl2anc 583	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝑖 + 0 ) = 𝑖)
34	33	fveq2d 6760	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(𝑖 + 0 )) = (𝐹‘𝑖))
35	31, 34	eqtr3d 2780	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘𝑖) ⨣ (𝐹‘ 0 )) = (𝐹‘𝑖))
36	25	oveq1d 7270	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘𝑖) ⨣ (𝐹‘ 0 )) = (𝑎 ⨣ (𝐹‘ 0 )))
37	35, 36, 25	3eqtr3d 2786	. . 3 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝑎 ⨣ (𝐹‘ 0 )) = 𝑎)
38	37, 29	r19.29a 3217	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → (𝑎 ⨣ (𝐹‘ 0 )) = 𝑎)
39	1, 2, 3, 12, 30, 38	ismgmid2 18267	1 ⊢ (𝜑 → (𝐹‘ 0 ) = (0g‘𝐻))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∃wrex 3064  ⟶wf 6414  –onto→wfo 6416  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  0_gc0g 17067  Mndcmnd 18300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fo 6424  df-fv 6426  df-riota 7212  df-ov 7258  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301

This theorem is referenced by:  mhmfmhm  18613  ghmgrp  18614