Theorem mhmid 13495

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 2206	. 2 ⊢ (0g‘𝐻) = (0g‘𝐻)
3		ghmgrp.q	. 2 ⊢ ⨣ = (+g‘𝐻)
4		ghmgrp.1	. . . 4 ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
5		fof 5505	. . . 4 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌)
6	4, 5	syl 14	. . 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 13306	. . . 4 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝑋)
11	7, 10	syl 14	. . 3 ⊢ (𝜑 → 0 ∈ 𝑋)
12	6, 11	ffvelcdmd 5723	. 2 ⊢ (𝜑 → (𝐹‘ 0 ) ∈ 𝑌)
13		simplll 533	. . . . . . 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 1283	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
16	7	ad3antrrr 492	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝐺 ∈ Mnd)
17	16, 10	syl 14	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 0 ∈ 𝑋)
18		simplr 528	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝑖 ∈ 𝑋)
19	15, 17, 18	mhmlem 13494	. . . . 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 13311	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝑖 ∈ 𝑋) → ( 0 + 𝑖) = 𝑖)
22	16, 18, 21	syl2anc 411	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ( 0 + 𝑖) = 𝑖)
23	22	fveq2d 5587	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘( 0 + 𝑖)) = (𝐹‘𝑖))
24	19, 23	eqtr3d 2241	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘ 0 ) ⨣ (𝐹‘𝑖)) = (𝐹‘𝑖))
25		simpr 110	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘𝑖) = 𝑎)
26	25	oveq2d 5967	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘ 0 ) ⨣ (𝐹‘𝑖)) = ((𝐹‘ 0 ) ⨣ 𝑎))
27	24, 26, 25	3eqtr3d 2247	. . 3 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘ 0 ) ⨣ 𝑎) = 𝑎)
28		foelcdmi 5638	. . . 4 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
29	4, 28	sylan 283	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
30	27, 29	r19.29a 2650	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → ((𝐹‘ 0 ) ⨣ 𝑎) = 𝑎)
31	15, 18, 17	mhmlem 13494	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(𝑖 + 0 )) = ((𝐹‘𝑖) ⨣ (𝐹‘ 0 )))
32	8, 20, 9	mndrid 13312	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝑖 ∈ 𝑋) → (𝑖 + 0 ) = 𝑖)
33	16, 18, 32	syl2anc 411	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝑖 + 0 ) = 𝑖)
34	33	fveq2d 5587	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(𝑖 + 0 )) = (𝐹‘𝑖))
35	31, 34	eqtr3d 2241	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘𝑖) ⨣ (𝐹‘ 0 )) = (𝐹‘𝑖))
36	25	oveq1d 5966	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘𝑖) ⨣ (𝐹‘ 0 )) = (𝑎 ⨣ (𝐹‘ 0 )))
37	35, 36, 25	3eqtr3d 2247	. . 3 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝑎 ⨣ (𝐹‘ 0 )) = 𝑎)
38	37, 29	r19.29a 2650	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → (𝑎 ⨣ (𝐹‘ 0 )) = 𝑎)
39	1, 2, 3, 12, 30, 38	ismgmid2 13256	1 ⊢ (𝜑 → (𝐹‘ 0 ) = (0g‘𝐻))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  ∃wrex 2486  ⟶wf 5272  –onto→wfo 5274  ‘cfv 5276  (class class class)co 5951  Basecbs 12876  +_gcplusg 12953  0_gc0g 13132  Mndcmnd 13292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-cnex 8023  ax-resscn 8024  ax-1re 8026  ax-addrcl 8029

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-fo 5282  df-fv 5284  df-riota 5906  df-ov 5954  df-inn 9044  df-2 9102  df-ndx 12879  df-slot 12880  df-base 12882  df-plusg 12966  df-0g 13134  df-mgm 13232  df-sgrp 13278  df-mnd 13293

This theorem is referenced by:  mhmfmhm  13497  ghmgrp  13498