Theorem mhmid 19088

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 2761	. 2 ⊢ (0g‘𝐻) = (0g‘𝐻)
3		ghmgrp.q	. 2 ⊢ ⨣ = (+g‘𝐻)
4		ghmgrp.1	. . . 4 ⊢ (𝜑 → 𝐹:𝑋–onto→𝑌)
5		fof 6774	. . . 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 18766	. . . 4 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝑋)
11	7, 10	syl 17	. . 3 ⊢ (𝜑 → 0 ∈ 𝑋)
12	6, 11	ffvelcdmd 7062	. 2 ⊢ (𝜑 → (𝐹‘ 0 ) ∈ 𝑌)
13		simplll 784	. . . . . . 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 1175	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
16	7	ad3antrrr 740	. . . . . . 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 778	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → 𝑖 ∈ 𝑋)
19	15, 17, 18	mhmlem 19087	. . . . 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 18771	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝑖 ∈ 𝑋) → ( 0 + 𝑖) = 𝑖)
22	16, 18, 21	syl2anc 593	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ( 0 + 𝑖) = 𝑖)
23	22	fveq2d 6867	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘( 0 + 𝑖)) = (𝐹‘𝑖))
24	19, 23	eqtr3d 2798	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘ 0 ) ⨣ (𝐹‘𝑖)) = (𝐹‘𝑖))
25		simpr 488	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘𝑖) = 𝑎)
26	25	oveq2d 7408	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘ 0 ) ⨣ (𝐹‘𝑖)) = ((𝐹‘ 0 ) ⨣ 𝑎))
27	24, 26, 25	3eqtr3d 2804	. . 3 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘ 0 ) ⨣ 𝑎) = 𝑎)
28		foelcdmi 6924	. . . 4 ⊢ ((𝐹:𝑋–onto→𝑌 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
29	4, 28	sylan 589	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → ∃𝑖 ∈ 𝑋 (𝐹‘𝑖) = 𝑎)
30	27, 29	r19.29a 3169	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → ((𝐹‘ 0 ) ⨣ 𝑎) = 𝑎)
31	15, 18, 17	mhmlem 19087	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(𝑖 + 0 )) = ((𝐹‘𝑖) ⨣ (𝐹‘ 0 )))
32	8, 20, 9	mndrid 18772	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝑖 ∈ 𝑋) → (𝑖 + 0 ) = 𝑖)
33	16, 18, 32	syl2anc 593	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝑖 + 0 ) = 𝑖)
34	33	fveq2d 6867	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝐹‘(𝑖 + 0 )) = (𝐹‘𝑖))
35	31, 34	eqtr3d 2798	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘𝑖) ⨣ (𝐹‘ 0 )) = (𝐹‘𝑖))
36	25	oveq1d 7407	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → ((𝐹‘𝑖) ⨣ (𝐹‘ 0 )) = (𝑎 ⨣ (𝐹‘ 0 )))
37	35, 36, 25	3eqtr3d 2804	. . 3 ⊢ ((((𝜑 ∧ 𝑎 ∈ 𝑌) ∧ 𝑖 ∈ 𝑋) ∧ (𝐹‘𝑖) = 𝑎) → (𝑎 ⨣ (𝐹‘ 0 )) = 𝑎)
38	37, 29	r19.29a 3169	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑌) → (𝑎 ⨣ (𝐹‘ 0 )) = 𝑎)
39	1, 2, 3, 12, 30, 38	ismgmid2 18685	1 ⊢ (𝜑 → (𝐹‘ 0 ) = (0g‘𝐻))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  ⟶wf 6513  –onto→wfo 6515  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  +_gcplusg 17269  0_gc0g 17451  Mndcmnd 18751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fo 6523  df-fv 6525  df-riota 7349  df-ov 7395  df-0g 17453  df-mgm 18657  df-sgrp 18736  df-mnd 18752

This theorem is referenced by:  mhmfmhm  19090  ghmgrp  19091