frgpupf - Metamath Proof Explorer

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

Theorem frgpupf 18386

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, 2-Oct-2015.)

**Hypotheses**
Ref	Expression
frgpup.b	⊢ 𝐵 = (Base‘𝐻)
frgpup.n	⊢ 𝑁 = (inv_g‘𝐻)
frgpup.t	⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))
frgpup.h	⊢ (𝜑 → 𝐻 ∈ Grp)
frgpup.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
frgpup.a	⊢ (𝜑 → 𝐹:𝐼⟶𝐵)
frgpup.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
frgpup.r	⊢ ∼ = ( ~_FG ‘𝐼)
frgpup.g	⊢ 𝐺 = (freeGrp‘𝐼)
frgpup.x	⊢ 𝑋 = (Base‘𝐺)
frgpup.e	⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ ⟨[𝑔] ∼ , (𝐻 Σ_g (𝑇 ∘ 𝑔))⟩)

**Assertion**
Ref	Expression
frgpupf	⊢ (𝜑 → 𝐸:𝑋⟶𝐵)

Distinct variable groups: 𝑦,𝑔,𝑧 𝑔,𝐻 𝑦,𝐹,𝑧 𝑦,𝑁,𝑧 𝐵,𝑔,𝑦,𝑧 𝑇,𝑔 ∼ ,𝑔 𝜑,𝑔,𝑦,𝑧 𝑦,𝐼,𝑧 𝑔,𝑊 Allowed substitution hints: ∼ (𝑦,𝑧) 𝑇(𝑦,𝑧) 𝐸(𝑦,𝑧,𝑔) 𝐹(𝑔) 𝐺(𝑦,𝑧,𝑔) 𝐻(𝑦,𝑧) 𝐼(𝑔) 𝑁(𝑔) 𝑉(𝑦,𝑧,𝑔) 𝑊(𝑦,𝑧) 𝑋(𝑦,𝑧,𝑔)

Proof of Theorem frgpupf Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frgpup.e	. . . 4 ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ ⟨[𝑔] ∼ , (𝐻 Σ_g (𝑇 ∘ 𝑔))⟩)
2		frgpup.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ Grp)
3		grpmnd 17630	. . . . . . 7 ⊢ (𝐻 ∈ Grp → 𝐻 ∈ Mnd)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Mnd)
5	4	adantr 472	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → 𝐻 ∈ Mnd)
6		frgpup.w	. . . . . . . 8 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
7		fviss 6418	. . . . . . . 8 ⊢ ( I ‘Word (𝐼 × 2_𝑜)) ⊆ Word (𝐼 × 2_𝑜)
8	6, 7	eqsstri 3776	. . . . . . 7 ⊢ 𝑊 ⊆ Word (𝐼 × 2_𝑜)
9	8	sseli 3740	. . . . . 6 ⊢ (𝑔 ∈ 𝑊 → 𝑔 ∈ Word (𝐼 × 2_𝑜))
10		frgpup.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐻)
11		frgpup.n	. . . . . . 7 ⊢ 𝑁 = (inv_g‘𝐻)
12		frgpup.t	. . . . . . 7 ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))
13		frgpup.i	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
14		frgpup.a	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)
15	10, 11, 12, 2, 13, 14	frgpuptf 18383	. . . . . 6 ⊢ (𝜑 → 𝑇:(𝐼 × 2_𝑜)⟶𝐵)
16		wrdco 13777	. . . . . 6 ⊢ ((𝑔 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑇:(𝐼 × 2_𝑜)⟶𝐵) → (𝑇 ∘ 𝑔) ∈ Word 𝐵)
17	9, 15, 16	syl2anr 496	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → (𝑇 ∘ 𝑔) ∈ Word 𝐵)
18	10	gsumwcl 17578	. . . . 5 ⊢ ((𝐻 ∈ Mnd ∧ (𝑇 ∘ 𝑔) ∈ Word 𝐵) → (𝐻 Σ_g (𝑇 ∘ 𝑔)) ∈ 𝐵)
19	5, 17, 18	syl2anc 696	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → (𝐻 Σ_g (𝑇 ∘ 𝑔)) ∈ 𝐵)
20		frgpup.r	. . . . . 6 ⊢ ∼ = ( ~_FG ‘𝐼)
21	6, 20	efger 18331	. . . . 5 ⊢ ∼ Er 𝑊
22	21	a1i 11	. . . 4 ⊢ (𝜑 → ∼ Er 𝑊)
23		fvex 6362	. . . . . 6 ⊢ ( I ‘Word (𝐼 × 2_𝑜)) ∈ V
24	6, 23	eqeltri 2835	. . . . 5 ⊢ 𝑊 ∈ V
25	24	a1i 11	. . . 4 ⊢ (𝜑 → 𝑊 ∈ V)
26		coeq2 5436	. . . . 5 ⊢ (𝑔 = ℎ → (𝑇 ∘ 𝑔) = (𝑇 ∘ ℎ))
27	26	oveq2d 6829	. . . 4 ⊢ (𝑔 = ℎ → (𝐻 Σ_g (𝑇 ∘ 𝑔)) = (𝐻 Σ_g (𝑇 ∘ ℎ)))
28	10, 11, 12, 2, 13, 14, 6, 20	frgpuplem 18385	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∼ ℎ) → (𝐻 Σ_g (𝑇 ∘ 𝑔)) = (𝐻 Σ_g (𝑇 ∘ ℎ)))
29	1, 19, 22, 25, 27, 28	qliftfund 8000	. . 3 ⊢ (𝜑 → Fun 𝐸)
30	1, 19, 22, 25	qliftf 8002	. . 3 ⊢ (𝜑 → (Fun 𝐸 ↔ 𝐸:(𝑊 / ∼ )⟶𝐵))
31	29, 30	mpbid 222	. 2 ⊢ (𝜑 → 𝐸:(𝑊 / ∼ )⟶𝐵)
32		frgpup.g	. . . . . . 7 ⊢ 𝐺 = (freeGrp‘𝐼)
33		eqid 2760	. . . . . . 7 ⊢ (freeMnd‘(𝐼 × 2_𝑜)) = (freeMnd‘(𝐼 × 2_𝑜))
34	32, 33, 20	frgpval 18371	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2_𝑜)) /_s ∼ ))
35	13, 34	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 = ((freeMnd‘(𝐼 × 2_𝑜)) /_s ∼ ))
36		2on 7737	. . . . . . . . 9 ⊢ 2_𝑜 ∈ On
37		xpexg 7125	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 2_𝑜 ∈ On) → (𝐼 × 2_𝑜) ∈ V)
38	13, 36, 37	sylancl 697	. . . . . . . 8 ⊢ (𝜑 → (𝐼 × 2_𝑜) ∈ V)
39		wrdexg 13501	. . . . . . . 8 ⊢ ((𝐼 × 2_𝑜) ∈ V → Word (𝐼 × 2_𝑜) ∈ V)
40		fvi 6417	. . . . . . . 8 ⊢ (Word (𝐼 × 2_𝑜) ∈ V → ( I ‘Word (𝐼 × 2_𝑜)) = Word (𝐼 × 2_𝑜))
41	38, 39, 40	3syl 18	. . . . . . 7 ⊢ (𝜑 → ( I ‘Word (𝐼 × 2_𝑜)) = Word (𝐼 × 2_𝑜))
42	6, 41	syl5eq 2806	. . . . . 6 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2_𝑜))
43		eqid 2760	. . . . . . . 8 ⊢ (Base‘(freeMnd‘(𝐼 × 2_𝑜))) = (Base‘(freeMnd‘(𝐼 × 2_𝑜)))
44	33, 43	frmdbas 17590	. . . . . . 7 ⊢ ((𝐼 × 2_𝑜) ∈ V → (Base‘(freeMnd‘(𝐼 × 2_𝑜))) = Word (𝐼 × 2_𝑜))
45	38, 44	syl 17	. . . . . 6 ⊢ (𝜑 → (Base‘(freeMnd‘(𝐼 × 2_𝑜))) = Word (𝐼 × 2_𝑜))
46	42, 45	eqtr4d 2797	. . . . 5 ⊢ (𝜑 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2_𝑜))))
47		fvex 6362	. . . . . . 7 ⊢ ( ~_FG ‘𝐼) ∈ V
48	20, 47	eqeltri 2835	. . . . . 6 ⊢ ∼ ∈ V
49	48	a1i 11	. . . . 5 ⊢ (𝜑 → ∼ ∈ V)
50		fvexd 6364	. . . . 5 ⊢ (𝜑 → (freeMnd‘(𝐼 × 2_𝑜)) ∈ V)
51	35, 46, 49, 50	qusbas 16407	. . . 4 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐺))
52		frgpup.x	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
53	51, 52	syl6reqr 2813	. . 3 ⊢ (𝜑 → 𝑋 = (𝑊 / ∼ ))
54	53	feq2d 6192	. 2 ⊢ (𝜑 → (𝐸:𝑋⟶𝐵 ↔ 𝐸:(𝑊 / ∼ )⟶𝐵))
55	31, 54	mpbird 247	1 ⊢ (𝜑 → 𝐸:𝑋⟶𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ∅c0 4058 ifcif 4230 ⟨cop 4327 ↦ cmpt 4881 I cid 5173 × cxp 5264 ran crn 5267 ∘ ccom 5270 Oncon0 5884 Fun wfun 6043 ⟶wf 6045 ‘cfv 6049 (class class class)co 6813 ↦ cmpt2 6815 2_𝑜c2o 7723 Er wer 7908 [cec 7909 / cqs 7910 Word cword 13477 Basecbs 16059 Σ_g cgsu 16303 /_s cqus 16367 Mndcmnd 17495 freeMndcfrmd 17585 Grpcgrp 17623 inv_gcminusg 17624 ~_FG cefg 18319 freeGrpcfrgp 18320

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-ot 4330 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-er 7911 df-ec 7913 df-qs 7917 df-map 8025 df-pm 8026 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-sup 8513 df-inf 8514 df-card 8955 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-dec 11686 df-uz 11880 df-fz 12520 df-fzo 12660 df-seq 12996 df-hash 13312 df-word 13485 df-concat 13487 df-s1 13488 df-substr 13489 df-splice 13490 df-s2 13793 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-sca 16159 df-vsca 16160 df-ip 16161 df-tset 16162 df-ple 16163 df-ds 16166 df-0g 16304 df-gsum 16305 df-imas 16370 df-qus 16371 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-submnd 17537 df-frmd 17587 df-grp 17626 df-minusg 17627 df-efg 18322 df-frgp 18323

This theorem is referenced by: frgpupval 18387 frgpup1 18388

W3C validator