frgpupf - Metamath Proof Explorer

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Theorem frgpupf 18186

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 17429	. . . . . . 7 ⊢ (𝐻 ∈ Grp → 𝐻 ∈ Mnd)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Mnd)
5	4	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → 𝐻 ∈ Mnd)
6		frgpup.w	. . . . . . . 8 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
7		fviss 6256	. . . . . . . 8 ⊢ ( I ‘Word (𝐼 × 2_𝑜)) ⊆ Word (𝐼 × 2_𝑜)
8	6, 7	eqsstri 3635	. . . . . . 7 ⊢ 𝑊 ⊆ Word (𝐼 × 2_𝑜)
9	8	sseli 3599	. . . . . 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 18183	. . . . . 6 ⊢ (𝜑 → 𝑇:(𝐼 × 2_𝑜)⟶𝐵)
16		wrdco 13577	. . . . . 6 ⊢ ((𝑔 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑇:(𝐼 × 2_𝑜)⟶𝐵) → (𝑇 ∘ 𝑔) ∈ Word 𝐵)
17	9, 15, 16	syl2anr 495	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → (𝑇 ∘ 𝑔) ∈ Word 𝐵)
18	10	gsumwcl 17377	. . . . 5 ⊢ ((𝐻 ∈ Mnd ∧ (𝑇 ∘ 𝑔) ∈ Word 𝐵) → (𝐻 Σ_g (𝑇 ∘ 𝑔)) ∈ 𝐵)
19	5, 17, 18	syl2anc 693	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → (𝐻 Σ_g (𝑇 ∘ 𝑔)) ∈ 𝐵)
20		frgpup.r	. . . . . 6 ⊢ ∼ = ( ~_FG ‘𝐼)
21	6, 20	efger 18131	. . . . 5 ⊢ ∼ Er 𝑊
22	21	a1i 11	. . . 4 ⊢ (𝜑 → ∼ Er 𝑊)
23		fvex 6201	. . . . . 6 ⊢ ( I ‘Word (𝐼 × 2_𝑜)) ∈ V
24	6, 23	eqeltri 2697	. . . . 5 ⊢ 𝑊 ∈ V
25	24	a1i 11	. . . 4 ⊢ (𝜑 → 𝑊 ∈ V)
26		coeq2 5280	. . . . 5 ⊢ (𝑔 = ℎ → (𝑇 ∘ 𝑔) = (𝑇 ∘ ℎ))
27	26	oveq2d 6666	. . . 4 ⊢ (𝑔 = ℎ → (𝐻 Σ_g (𝑇 ∘ 𝑔)) = (𝐻 Σ_g (𝑇 ∘ ℎ)))
28	10, 11, 12, 2, 13, 14, 6, 20	frgpuplem 18185	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∼ ℎ) → (𝐻 Σ_g (𝑇 ∘ 𝑔)) = (𝐻 Σ_g (𝑇 ∘ ℎ)))
29	1, 19, 22, 25, 27, 28	qliftfund 7833	. . 3 ⊢ (𝜑 → Fun 𝐸)
30	1, 19, 22, 25	qliftf 7835	. . 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 2622	. . . . . . 7 ⊢ (freeMnd‘(𝐼 × 2_𝑜)) = (freeMnd‘(𝐼 × 2_𝑜))
34	32, 33, 20	frgpval 18171	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2_𝑜)) /_s ∼ ))
35	13, 34	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 = ((freeMnd‘(𝐼 × 2_𝑜)) /_s ∼ ))
36		2on 7568	. . . . . . . . 9 ⊢ 2_𝑜 ∈ On
37		xpexg 6960	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 2_𝑜 ∈ On) → (𝐼 × 2_𝑜) ∈ V)
38	13, 36, 37	sylancl 694	. . . . . . . 8 ⊢ (𝜑 → (𝐼 × 2_𝑜) ∈ V)
39		wrdexg 13315	. . . . . . . 8 ⊢ ((𝐼 × 2_𝑜) ∈ V → Word (𝐼 × 2_𝑜) ∈ V)
40		fvi 6255	. . . . . . . 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 2668	. . . . . 6 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2_𝑜))
43		eqid 2622	. . . . . . . 8 ⊢ (Base‘(freeMnd‘(𝐼 × 2_𝑜))) = (Base‘(freeMnd‘(𝐼 × 2_𝑜)))
44	33, 43	frmdbas 17389	. . . . . . 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 2659	. . . . 5 ⊢ (𝜑 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2_𝑜))))
47		fvex 6201	. . . . . . 7 ⊢ ( ~_FG ‘𝐼) ∈ V
48	20, 47	eqeltri 2697	. . . . . 6 ⊢ ∼ ∈ V
49	48	a1i 11	. . . . 5 ⊢ (𝜑 → ∼ ∈ V)
50		fvexd 6203	. . . . 5 ⊢ (𝜑 → (freeMnd‘(𝐼 × 2_𝑜)) ∈ V)
51	35, 46, 49, 50	qusbas 16205	. . . 4 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐺))
52		frgpup.x	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
53	51, 52	syl6reqr 2675	. . 3 ⊢ (𝜑 → 𝑋 = (𝑊 / ∼ ))
54	53	feq2d 6031	. 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 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 ifcif 4086 ⟨cop 4183 ↦ cmpt 4729 I cid 5023 × cxp 5112 ran crn 5115 ∘ ccom 5118 Oncon0 5723 Fun wfun 5882 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 2_𝑜c2o 7554 Er wer 7739 [cec 7740 / cqs 7741 Word cword 13291 Basecbs 15857 Σ_g cgsu 16101 /_s cqus 16165 Mndcmnd 17294 freeMndcfrmd 17384 Grpcgrp 17422 inv_gcminusg 17423 ~_FG cefg 18119 freeGrpcfrgp 18120

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-s2 13593 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-0g 16102 df-gsum 16103 df-imas 16168 df-qus 16169 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-frmd 17386 df-grp 17425 df-minusg 17426 df-efg 18122 df-frgp 18123

This theorem is referenced by: frgpupval 18187 frgpup1 18188

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