frmdup1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > frmdup1		Structured version Visualization version GIF version

Theorem frmdup1 18807

Description: Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 27-Sep-2015.)

**Hypotheses**
Ref	Expression
frmdup.m	⊢ 𝑀 = (freeMnd‘𝐼)
frmdup.b	⊢ 𝐵 = (Base‘𝐺)
frmdup.e	⊢ 𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σ_g (𝐴 ∘ 𝑥)))
frmdup.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
frmdup.i	⊢ (𝜑 → 𝐼 ∈ 𝑋)
frmdup.a	⊢ (𝜑 → 𝐴:𝐼⟶𝐵)

**Assertion**
Ref	Expression
frmdup1	⊢ (𝜑 → 𝐸 ∈ (𝑀 MndHom 𝐺))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐺 𝜑,𝑥 𝑥,𝐼 Allowed substitution hints: 𝐸(𝑥) 𝑀(𝑥) 𝑋(𝑥)

Proof of Theorem frmdup1 Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frmdup.i	. . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑋)
2		frmdup.m	. . . 4 ⊢ 𝑀 = (freeMnd‘𝐼)
3	2	frmdmnd 18802	. . 3 ⊢ (𝐼 ∈ 𝑋 → 𝑀 ∈ Mnd)
4	1, 3	syl 17	. 2 ⊢ (𝜑 → 𝑀 ∈ Mnd)
5		frmdup.g	. 2 ⊢ (𝜑 → 𝐺 ∈ Mnd)
6	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐼) → 𝐺 ∈ Mnd)
7		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐼) → 𝑥 ∈ Word 𝐼)
8		frmdup.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴:𝐼⟶𝐵)
9	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐼) → 𝐴:𝐼⟶𝐵)
10		wrdco 14806	. . . . . . 7 ⊢ ((𝑥 ∈ Word 𝐼 ∧ 𝐴:𝐼⟶𝐵) → (𝐴 ∘ 𝑥) ∈ Word 𝐵)
11	7, 9, 10	syl2anc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐼) → (𝐴 ∘ 𝑥) ∈ Word 𝐵)
12		frmdup.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
13	12	gsumwcl 18782	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∘ 𝑥) ∈ Word 𝐵) → (𝐺 Σ_g (𝐴 ∘ 𝑥)) ∈ 𝐵)
14	6, 11, 13	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ Word 𝐼) → (𝐺 Σ_g (𝐴 ∘ 𝑥)) ∈ 𝐵)
15		frmdup.e	. . . . 5 ⊢ 𝐸 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σ_g (𝐴 ∘ 𝑥)))
16	14, 15	fmptd 7118	. . . 4 ⊢ (𝜑 → 𝐸:Word 𝐼⟶𝐵)
17		eqid 2727	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
18	2, 17	frmdbas 18795	. . . . . 6 ⊢ (𝐼 ∈ 𝑋 → (Base‘𝑀) = Word 𝐼)
19	1, 18	syl 17	. . . . 5 ⊢ (𝜑 → (Base‘𝑀) = Word 𝐼)
20	19	feq2d 6702	. . . 4 ⊢ (𝜑 → (𝐸:(Base‘𝑀)⟶𝐵 ↔ 𝐸:Word 𝐼⟶𝐵))
21	16, 20	mpbird 257	. . 3 ⊢ (𝜑 → 𝐸:(Base‘𝑀)⟶𝐵)
22	2, 17	frmdelbas 18796	. . . . . . . . 9 ⊢ (𝑦 ∈ (Base‘𝑀) → 𝑦 ∈ Word 𝐼)
23	22	ad2antrl 727	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑦 ∈ Word 𝐼)
24	2, 17	frmdelbas 18796	. . . . . . . . 9 ⊢ (𝑧 ∈ (Base‘𝑀) → 𝑧 ∈ Word 𝐼)
25	24	ad2antll 728	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝑧 ∈ Word 𝐼)
26	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝐴:𝐼⟶𝐵)
27		ccatco 14810	. . . . . . . 8 ⊢ ((𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼 ∧ 𝐴:𝐼⟶𝐵) → (𝐴 ∘ (𝑦 ++ 𝑧)) = ((𝐴 ∘ 𝑦) ++ (𝐴 ∘ 𝑧)))
28	23, 25, 26, 27	syl3anc 1369	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴 ∘ (𝑦 ++ 𝑧)) = ((𝐴 ∘ 𝑦) ++ (𝐴 ∘ 𝑧)))
29	28	oveq2d 7430	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σ_g (𝐴 ∘ (𝑦 ++ 𝑧))) = (𝐺 Σ_g ((𝐴 ∘ 𝑦) ++ (𝐴 ∘ 𝑧))))
30	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → 𝐺 ∈ Mnd)
31		wrdco 14806	. . . . . . . 8 ⊢ ((𝑦 ∈ Word 𝐼 ∧ 𝐴:𝐼⟶𝐵) → (𝐴 ∘ 𝑦) ∈ Word 𝐵)
32	23, 26, 31	syl2anc 583	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴 ∘ 𝑦) ∈ Word 𝐵)
33		wrdco 14806	. . . . . . . 8 ⊢ ((𝑧 ∈ Word 𝐼 ∧ 𝐴:𝐼⟶𝐵) → (𝐴 ∘ 𝑧) ∈ Word 𝐵)
34	25, 26, 33	syl2anc 583	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐴 ∘ 𝑧) ∈ Word 𝐵)
35		eqid 2727	. . . . . . . 8 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
36	12, 35	gsumccat 18784	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ (𝐴 ∘ 𝑦) ∈ Word 𝐵 ∧ (𝐴 ∘ 𝑧) ∈ Word 𝐵) → (𝐺 Σ_g ((𝐴 ∘ 𝑦) ++ (𝐴 ∘ 𝑧))) = ((𝐺 Σ_g (𝐴 ∘ 𝑦))(+_g‘𝐺)(𝐺 Σ_g (𝐴 ∘ 𝑧))))
37	30, 32, 34, 36	syl3anc 1369	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σ_g ((𝐴 ∘ 𝑦) ++ (𝐴 ∘ 𝑧))) = ((𝐺 Σ_g (𝐴 ∘ 𝑦))(+_g‘𝐺)(𝐺 Σ_g (𝐴 ∘ 𝑧))))
38	29, 37	eqtrd 2767	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐺 Σ_g (𝐴 ∘ (𝑦 ++ 𝑧))) = ((𝐺 Σ_g (𝐴 ∘ 𝑦))(+_g‘𝐺)(𝐺 Σ_g (𝐴 ∘ 𝑧))))
39		eqid 2727	. . . . . . . . 9 ⊢ (+_g‘𝑀) = (+_g‘𝑀)
40	2, 17, 39	frmdadd 18798	. . . . . . . 8 ⊢ ((𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀)) → (𝑦(+_g‘𝑀)𝑧) = (𝑦 ++ 𝑧))
41	40	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦(+_g‘𝑀)𝑧) = (𝑦 ++ 𝑧))
42	41	fveq2d 6895	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+_g‘𝑀)𝑧)) = (𝐸‘(𝑦 ++ 𝑧)))
43		ccatcl 14548	. . . . . . . 8 ⊢ ((𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
44	23, 25, 43	syl2anc 583	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝑦 ++ 𝑧) ∈ Word 𝐼)
45		coeq2 5855	. . . . . . . . 9 ⊢ (𝑥 = (𝑦 ++ 𝑧) → (𝐴 ∘ 𝑥) = (𝐴 ∘ (𝑦 ++ 𝑧)))
46	45	oveq2d 7430	. . . . . . . 8 ⊢ (𝑥 = (𝑦 ++ 𝑧) → (𝐺 Σ_g (𝐴 ∘ 𝑥)) = (𝐺 Σ_g (𝐴 ∘ (𝑦 ++ 𝑧))))
47		ovex 7447	. . . . . . . 8 ⊢ (𝐺 Σ_g (𝐴 ∘ 𝑥)) ∈ V
48	46, 15, 47	fvmpt3i 7004	. . . . . . 7 ⊢ ((𝑦 ++ 𝑧) ∈ Word 𝐼 → (𝐸‘(𝑦 ++ 𝑧)) = (𝐺 Σ_g (𝐴 ∘ (𝑦 ++ 𝑧))))
49	44, 48	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦 ++ 𝑧)) = (𝐺 Σ_g (𝐴 ∘ (𝑦 ++ 𝑧))))
50	42, 49	eqtrd 2767	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+_g‘𝑀)𝑧)) = (𝐺 Σ_g (𝐴 ∘ (𝑦 ++ 𝑧))))
51		coeq2 5855	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 ∘ 𝑥) = (𝐴 ∘ 𝑦))
52	51	oveq2d 7430	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐺 Σ_g (𝐴 ∘ 𝑥)) = (𝐺 Σ_g (𝐴 ∘ 𝑦)))
53	52, 15, 47	fvmpt3i 7004	. . . . . . 7 ⊢ (𝑦 ∈ Word 𝐼 → (𝐸‘𝑦) = (𝐺 Σ_g (𝐴 ∘ 𝑦)))
54		coeq2 5855	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝐴 ∘ 𝑥) = (𝐴 ∘ 𝑧))
55	54	oveq2d 7430	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝐺 Σ_g (𝐴 ∘ 𝑥)) = (𝐺 Σ_g (𝐴 ∘ 𝑧)))
56	55, 15, 47	fvmpt3i 7004	. . . . . . 7 ⊢ (𝑧 ∈ Word 𝐼 → (𝐸‘𝑧) = (𝐺 Σ_g (𝐴 ∘ 𝑧)))
57	53, 56	oveqan12d 7433	. . . . . 6 ⊢ ((𝑦 ∈ Word 𝐼 ∧ 𝑧 ∈ Word 𝐼) → ((𝐸‘𝑦)(+_g‘𝐺)(𝐸‘𝑧)) = ((𝐺 Σ_g (𝐴 ∘ 𝑦))(+_g‘𝐺)(𝐺 Σ_g (𝐴 ∘ 𝑧))))
58	23, 25, 57	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → ((𝐸‘𝑦)(+_g‘𝐺)(𝐸‘𝑧)) = ((𝐺 Σ_g (𝐴 ∘ 𝑦))(+_g‘𝐺)(𝐺 Σ_g (𝐴 ∘ 𝑧))))
59	38, 50, 58	3eqtr4d 2777	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ (Base‘𝑀) ∧ 𝑧 ∈ (Base‘𝑀))) → (𝐸‘(𝑦(+_g‘𝑀)𝑧)) = ((𝐸‘𝑦)(+_g‘𝐺)(𝐸‘𝑧)))
60	59	ralrimivva 3195	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+_g‘𝑀)𝑧)) = ((𝐸‘𝑦)(+_g‘𝐺)(𝐸‘𝑧)))
61		wrd0 14513	. . . 4 ⊢ ∅ ∈ Word 𝐼
62		coeq2 5855	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝐴 ∘ 𝑥) = (𝐴 ∘ ∅))
63		co02 6258	. . . . . . . 8 ⊢ (𝐴 ∘ ∅) = ∅
64	62, 63	eqtrdi 2783	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝐴 ∘ 𝑥) = ∅)
65	64	oveq2d 7430	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐺 Σ_g (𝐴 ∘ 𝑥)) = (𝐺 Σ_g ∅))
66		eqid 2727	. . . . . . 7 ⊢ (0_g‘𝐺) = (0_g‘𝐺)
67	66	gsum0 18635	. . . . . 6 ⊢ (𝐺 Σ_g ∅) = (0_g‘𝐺)
68	65, 67	eqtrdi 2783	. . . . 5 ⊢ (𝑥 = ∅ → (𝐺 Σ_g (𝐴 ∘ 𝑥)) = (0_g‘𝐺))
69	68, 15, 47	fvmpt3i 7004	. . . 4 ⊢ (∅ ∈ Word 𝐼 → (𝐸‘∅) = (0_g‘𝐺))
70	61, 69	mp1i 13	. . 3 ⊢ (𝜑 → (𝐸‘∅) = (0_g‘𝐺))
71	21, 60, 70	3jca 1126	. 2 ⊢ (𝜑 → (𝐸:(Base‘𝑀)⟶𝐵 ∧ ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+_g‘𝑀)𝑧)) = ((𝐸‘𝑦)(+_g‘𝐺)(𝐸‘𝑧)) ∧ (𝐸‘∅) = (0_g‘𝐺)))
72	2	frmd0 18803	. . 3 ⊢ ∅ = (0_g‘𝑀)
73	17, 12, 39, 35, 72, 66	ismhm 18733	. 2 ⊢ (𝐸 ∈ (𝑀 MndHom 𝐺) ↔ ((𝑀 ∈ Mnd ∧ 𝐺 ∈ Mnd) ∧ (𝐸:(Base‘𝑀)⟶𝐵 ∧ ∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)(𝐸‘(𝑦(+_g‘𝑀)𝑧)) = ((𝐸‘𝑦)(+_g‘𝐺)(𝐸‘𝑧)) ∧ (𝐸‘∅) = (0_g‘𝐺))))
74	4, 5, 71, 73	syl21anbrc 1342	1 ⊢ (𝜑 → 𝐸 ∈ (𝑀 MndHom 𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ∅c0 4318 ↦ cmpt 5225 ∘ ccom 5676 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 Word cword 14488 ++ cconcat 14544 Basecbs 17171 +_gcplusg 17224 0_gc0g 17412 Σ_g cgsu 17413 Mndcmnd 18685 MndHom cmhm 18729 freeMndcfrmd 18790

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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-card 9954 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-n0 12495 df-z 12581 df-uz 12845 df-fz 13509 df-fzo 13652 df-seq 13991 df-hash 14314 df-word 14489 df-concat 14545 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-0g 17414 df-gsum 17415 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-submnd 18732 df-frmd 18792

This theorem is referenced by: frmdup3 18810

W3C validator