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 18009

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 17252	. . . . . . 7 ⊢ (𝐻 ∈ Grp → 𝐻 ∈ Mnd)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Mnd)
5	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → 𝐻 ∈ Mnd)
6		frgpup.w	. . . . . . . 8 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
7		fviss 6166	. . . . . . . 8 ⊢ ( I ‘Word (𝐼 × 2_𝑜)) ⊆ Word (𝐼 × 2_𝑜)
8	6, 7	eqsstri 3598	. . . . . . 7 ⊢ 𝑊 ⊆ Word (𝐼 × 2_𝑜)
9	8	sseli 3564	. . . . . 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 18006	. . . . . 6 ⊢ (𝜑 → 𝑇:(𝐼 × 2_𝑜)⟶𝐵)
16		wrdco 13428	. . . . . 6 ⊢ ((𝑔 ∈ Word (𝐼 × 2_𝑜) ∧ 𝑇:(𝐼 × 2_𝑜)⟶𝐵) → (𝑇 ∘ 𝑔) ∈ Word 𝐵)
17	9, 15, 16	syl2anr 494	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → (𝑇 ∘ 𝑔) ∈ Word 𝐵)
18	10	gsumwcl 17200	. . . . 5 ⊢ ((𝐻 ∈ Mnd ∧ (𝑇 ∘ 𝑔) ∈ Word 𝐵) → (𝐻 Σ_g (𝑇 ∘ 𝑔)) ∈ 𝐵)
19	5, 17, 18	syl2anc 691	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → (𝐻 Σ_g (𝑇 ∘ 𝑔)) ∈ 𝐵)
20		frgpup.r	. . . . . 6 ⊢ ∼ = ( ~_FG ‘𝐼)
21	6, 20	efger 17954	. . . . 5 ⊢ ∼ Er 𝑊
22	21	a1i 11	. . . 4 ⊢ (𝜑 → ∼ Er 𝑊)
23		fvex 6113	. . . . . 6 ⊢ ( I ‘Word (𝐼 × 2_𝑜)) ∈ V
24	6, 23	eqeltri 2684	. . . . 5 ⊢ 𝑊 ∈ V
25	24	a1i 11	. . . 4 ⊢ (𝜑 → 𝑊 ∈ V)
26		coeq2 5202	. . . . 5 ⊢ (𝑔 = ℎ → (𝑇 ∘ 𝑔) = (𝑇 ∘ ℎ))
27	26	oveq2d 6565	. . . 4 ⊢ (𝑔 = ℎ → (𝐻 Σ_g (𝑇 ∘ 𝑔)) = (𝐻 Σ_g (𝑇 ∘ ℎ)))
28	10, 11, 12, 2, 13, 14, 6, 20	frgpuplem 18008	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∼ ℎ) → (𝐻 Σ_g (𝑇 ∘ 𝑔)) = (𝐻 Σ_g (𝑇 ∘ ℎ)))
29	1, 19, 22, 25, 27, 28	qliftfund 7720	. . 3 ⊢ (𝜑 → Fun 𝐸)
30	1, 19, 22, 25	qliftf 7722	. . 3 ⊢ (𝜑 → (Fun 𝐸 ↔ 𝐸:(𝑊 / ∼ )⟶𝐵))
31	29, 30	mpbid 221	. 2 ⊢ (𝜑 → 𝐸:(𝑊 / ∼ )⟶𝐵)
32		frgpup.g	. . . . . . 7 ⊢ 𝐺 = (freeGrp‘𝐼)
33		eqid 2610	. . . . . . 7 ⊢ (freeMnd‘(𝐼 × 2_𝑜)) = (freeMnd‘(𝐼 × 2_𝑜))
34	32, 33, 20	frgpval 17994	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2_𝑜)) /_s ∼ ))
35	13, 34	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 = ((freeMnd‘(𝐼 × 2_𝑜)) /_s ∼ ))
36		2on 7455	. . . . . . . . 9 ⊢ 2_𝑜 ∈ On
37		xpexg 6858	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 2_𝑜 ∈ On) → (𝐼 × 2_𝑜) ∈ V)
38	13, 36, 37	sylancl 693	. . . . . . . 8 ⊢ (𝜑 → (𝐼 × 2_𝑜) ∈ V)
39		wrdexg 13170	. . . . . . . 8 ⊢ ((𝐼 × 2_𝑜) ∈ V → Word (𝐼 × 2_𝑜) ∈ V)
40		fvi 6165	. . . . . . . 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 2656	. . . . . 6 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2_𝑜))
43		eqid 2610	. . . . . . . 8 ⊢ (Base‘(freeMnd‘(𝐼 × 2_𝑜))) = (Base‘(freeMnd‘(𝐼 × 2_𝑜)))
44	33, 43	frmdbas 17212	. . . . . . 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 2647	. . . . 5 ⊢ (𝜑 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2_𝑜))))
47		fvex 6113	. . . . . . 7 ⊢ ( ~_FG ‘𝐼) ∈ V
48	20, 47	eqeltri 2684	. . . . . 6 ⊢ ∼ ∈ V
49	48	a1i 11	. . . . 5 ⊢ (𝜑 → ∼ ∈ V)
50		fvex 6113	. . . . . 6 ⊢ (freeMnd‘(𝐼 × 2_𝑜)) ∈ V
51	50	a1i 11	. . . . 5 ⊢ (𝜑 → (freeMnd‘(𝐼 × 2_𝑜)) ∈ V)
52	35, 46, 49, 51	qusbas 16028	. . . 4 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐺))
53		frgpup.x	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
54	52, 53	syl6reqr 2663	. . 3 ⊢ (𝜑 → 𝑋 = (𝑊 / ∼ ))
55	54	feq2d 5944	. 2 ⊢ (𝜑 → (𝐸:𝑋⟶𝐵 ↔ 𝐸:(𝑊 / ∼ )⟶𝐵))
56	31, 55	mpbird 246	1 ⊢ (𝜑 → 𝐸:𝑋⟶𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ifcif 4036 ⟨cop 4131 ↦ cmpt 4643 I cid 4948 × cxp 5036 ran crn 5039 ∘ ccom 5042 Oncon0 5640 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 2_𝑜c2o 7441 Er wer 7626 [cec 7627 / cqs 7628 Word cword 13146 Basecbs 15695 Σ_g cgsu 15924 /_s cqus 15988 Mndcmnd 17117 freeMndcfrmd 17207 Grpcgrp 17245 inv_gcminusg 17246 ~_FG cefg 17942 freeGrpcfrgp 17943

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-ot 4134 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-substr 13158 df-splice 13159 df-s2 13444 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-0g 15925 df-gsum 15926 df-imas 15991 df-qus 15992 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-frmd 17209 df-grp 17248 df-minusg 17249 df-efg 17945 df-frgp 17946

This theorem is referenced by: frgpupval 18010 frgpup1 18011

W3C validator