frgpup2 - Metamath Proof Explorer

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Theorem frgpup2 18189

Description: The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)

**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))⟩)
frgpup.u	⊢ 𝑈 = (var_FGrp‘𝐼)
frgpup.y	⊢ (𝜑 → 𝐴 ∈ 𝐼)

**Assertion**
Ref	Expression
frgpup2	⊢ (𝜑 → (𝐸‘(𝑈‘𝐴)) = (𝐹‘𝐴))

Distinct variable groups: 𝑦,𝑔,𝑧,𝐴 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) 𝑈(𝑦,𝑧,𝑔) 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 frgpup2**
Step	Hyp	Ref	Expression
1		frgpup.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
2		frgpup.y	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐼)
3		frgpup.r	. . . . 5 ⊢ ∼ = ( ~_FG ‘𝐼)
4		frgpup.u	. . . . 5 ⊢ 𝑈 = (var_FGrp‘𝐼)
5	3, 4	vrgpval 18180	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
6	1, 2, 5	syl2anc 693	. . 3 ⊢ (𝜑 → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
7	6	fveq2d 6195	. 2 ⊢ (𝜑 → (𝐸‘(𝑈‘𝐴)) = (𝐸‘[⟨“⟨𝐴, ∅⟩”⟩] ∼ ))
8		0ex 4790	. . . . . . . 8 ⊢ ∅ ∈ V
9	8	prid1 4297	. . . . . . 7 ⊢ ∅ ∈ {∅, 1_𝑜}
10		df2o3 7573	. . . . . . 7 ⊢ 2_𝑜 = {∅, 1_𝑜}
11	9, 10	eleqtrri 2700	. . . . . 6 ⊢ ∅ ∈ 2_𝑜
12		opelxpi 5148	. . . . . 6 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2_𝑜) → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2_𝑜))
13	2, 11, 12	sylancl 694	. . . . 5 ⊢ (𝜑 → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2_𝑜))
14	13	s1cld 13383	. . . 4 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ Word (𝐼 × 2_𝑜))
15		frgpup.w	. . . . 5 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_𝑜))
16		2on 7568	. . . . . . 7 ⊢ 2_𝑜 ∈ On
17		xpexg 6960	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 2_𝑜 ∈ On) → (𝐼 × 2_𝑜) ∈ V)
18	1, 16, 17	sylancl 694	. . . . . 6 ⊢ (𝜑 → (𝐼 × 2_𝑜) ∈ V)
19		wrdexg 13315	. . . . . 6 ⊢ ((𝐼 × 2_𝑜) ∈ V → Word (𝐼 × 2_𝑜) ∈ V)
20		fvi 6255	. . . . . 6 ⊢ (Word (𝐼 × 2_𝑜) ∈ V → ( I ‘Word (𝐼 × 2_𝑜)) = Word (𝐼 × 2_𝑜))
21	18, 19, 20	3syl 18	. . . . 5 ⊢ (𝜑 → ( I ‘Word (𝐼 × 2_𝑜)) = Word (𝐼 × 2_𝑜))
22	15, 21	syl5eq 2668	. . . 4 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2_𝑜))
23	14, 22	eleqtrrd 2704	. . 3 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊)
24		frgpup.b	. . . 4 ⊢ 𝐵 = (Base‘𝐻)
25		frgpup.n	. . . 4 ⊢ 𝑁 = (inv_g‘𝐻)
26		frgpup.t	. . . 4 ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))))
27		frgpup.h	. . . 4 ⊢ (𝜑 → 𝐻 ∈ Grp)
28		frgpup.a	. . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵)
29		frgpup.g	. . . 4 ⊢ 𝐺 = (freeGrp‘𝐼)
30		frgpup.x	. . . 4 ⊢ 𝑋 = (Base‘𝐺)
31		frgpup.e	. . . 4 ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ ⟨[𝑔] ∼ , (𝐻 Σ_g (𝑇 ∘ 𝑔))⟩)
32	24, 25, 26, 27, 1, 28, 15, 3, 29, 30, 31	frgpupval 18187	. . 3 ⊢ ((𝜑 ∧ ⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊) → (𝐸‘[⟨“⟨𝐴, ∅⟩”⟩] ∼ ) = (𝐻 Σ_g (𝑇 ∘ ⟨“⟨𝐴, ∅⟩”⟩)))
33	23, 32	mpdan 702	. 2 ⊢ (𝜑 → (𝐸‘[⟨“⟨𝐴, ∅⟩”⟩] ∼ ) = (𝐻 Σ_g (𝑇 ∘ ⟨“⟨𝐴, ∅⟩”⟩)))
34	24, 25, 26, 27, 1, 28	frgpuptf 18183	. . . . . 6 ⊢ (𝜑 → 𝑇:(𝐼 × 2_𝑜)⟶𝐵)
35		s1co 13579	. . . . . 6 ⊢ ((⟨𝐴, ∅⟩ ∈ (𝐼 × 2_𝑜) ∧ 𝑇:(𝐼 × 2_𝑜)⟶𝐵) → (𝑇 ∘ ⟨“⟨𝐴, ∅⟩”⟩) = ⟨“(𝑇‘⟨𝐴, ∅⟩)”⟩)
36	13, 34, 35	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝑇 ∘ ⟨“⟨𝐴, ∅⟩”⟩) = ⟨“(𝑇‘⟨𝐴, ∅⟩)”⟩)
37		df-ov 6653	. . . . . . 7 ⊢ (𝐴𝑇∅) = (𝑇‘⟨𝐴, ∅⟩)
38		iftrue 4092	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = (𝐹‘𝑦))
39		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴))
40	38, 39	sylan9eqr 2678	. . . . . . . . 9 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = ∅) → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = (𝐹‘𝐴))
41		fvex 6201	. . . . . . . . 9 ⊢ (𝐹‘𝐴) ∈ V
42	40, 26, 41	ovmpt2a 6791	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2_𝑜) → (𝐴𝑇∅) = (𝐹‘𝐴))
43	2, 11, 42	sylancl 694	. . . . . . 7 ⊢ (𝜑 → (𝐴𝑇∅) = (𝐹‘𝐴))
44	37, 43	syl5eqr 2670	. . . . . 6 ⊢ (𝜑 → (𝑇‘⟨𝐴, ∅⟩) = (𝐹‘𝐴))
45	44	s1eqd 13381	. . . . 5 ⊢ (𝜑 → ⟨“(𝑇‘⟨𝐴, ∅⟩)”⟩ = ⟨“(𝐹‘𝐴)”⟩)
46	36, 45	eqtrd 2656	. . . 4 ⊢ (𝜑 → (𝑇 ∘ ⟨“⟨𝐴, ∅⟩”⟩) = ⟨“(𝐹‘𝐴)”⟩)
47	46	oveq2d 6666	. . 3 ⊢ (𝜑 → (𝐻 Σ_g (𝑇 ∘ ⟨“⟨𝐴, ∅⟩”⟩)) = (𝐻 Σ_g ⟨“(𝐹‘𝐴)”⟩))
48	28, 2	ffvelrnd 6360	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝐵)
49	24	gsumws1 17376	. . . 4 ⊢ ((𝐹‘𝐴) ∈ 𝐵 → (𝐻 Σ_g ⟨“(𝐹‘𝐴)”⟩) = (𝐹‘𝐴))
50	48, 49	syl 17	. . 3 ⊢ (𝜑 → (𝐻 Σ_g ⟨“(𝐹‘𝐴)”⟩) = (𝐹‘𝐴))
51	47, 50	eqtrd 2656	. 2 ⊢ (𝜑 → (𝐻 Σ_g (𝑇 ∘ ⟨“⟨𝐴, ∅⟩”⟩)) = (𝐹‘𝐴))
52	7, 33, 51	3eqtrd 2660	1 ⊢ (𝜑 → (𝐸‘(𝑈‘𝐴)) = (𝐹‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 ifcif 4086 {cpr 4179 ⟨cop 4183 ↦ cmpt 4729 I cid 5023 × cxp 5112 ran crn 5115 ∘ ccom 5118 Oncon0 5723 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1_𝑜c1o 7553 2_𝑜c2o 7554 [cec 7740 Word cword 13291 ⟨“cs1 13294 Basecbs 15857 Σ_g cgsu 16101 Grpcgrp 17422 inv_gcminusg 17423 ~_FG cefg 18119 freeGrpcfrgp 18120 var_FGrpcvrgp 18121

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 df-vrgp 18124

This theorem is referenced by: frgpup3 18191

W3C validator