frgpup2 - Metamath Proof Explorer

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Theorem frgpup2 18662

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 18653	. . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
6	1, 2, 5	syl2anc 576	. . 3 ⊢ (𝜑 → (𝑈‘𝐴) = [⟨“⟨𝐴, ∅⟩”⟩] ∼ )
7	6	fveq2d 6503	. 2 ⊢ (𝜑 → (𝐸‘(𝑈‘𝐴)) = (𝐸‘[⟨“⟨𝐴, ∅⟩”⟩] ∼ ))
8		0ex 5068	. . . . . . . 8 ⊢ ∅ ∈ V
9	8	prid1 4572	. . . . . . 7 ⊢ ∅ ∈ {∅, 1_o}
10		df2o3 7919	. . . . . . 7 ⊢ 2_o = {∅, 1_o}
11	9, 10	eleqtrri 2866	. . . . . 6 ⊢ ∅ ∈ 2_o
12		opelxpi 5444	. . . . . 6 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2_o) → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2_o))
13	2, 11, 12	sylancl 577	. . . . 5 ⊢ (𝜑 → ⟨𝐴, ∅⟩ ∈ (𝐼 × 2_o))
14	13	s1cld 13766	. . . 4 ⊢ (𝜑 → ⟨“⟨𝐴, ∅⟩”⟩ ∈ Word (𝐼 × 2_o))
15		frgpup.w	. . . . 5 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o))
16		2on 7914	. . . . . . 7 ⊢ 2_o ∈ On
17		xpexg 7290	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 2_o ∈ On) → (𝐼 × 2_o) ∈ V)
18	1, 16, 17	sylancl 577	. . . . . 6 ⊢ (𝜑 → (𝐼 × 2_o) ∈ V)
19		wrdexg 13682	. . . . . 6 ⊢ ((𝐼 × 2_o) ∈ V → Word (𝐼 × 2_o) ∈ V)
20		fvi 6568	. . . . . 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 2827	. . . 4 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2_o))
23	14, 22	eleqtrrd 2870	. . 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 18660	. . 3 ⊢ ((𝜑 ∧ ⟨“⟨𝐴, ∅⟩”⟩ ∈ 𝑊) → (𝐸‘[⟨“⟨𝐴, ∅⟩”⟩] ∼ ) = (𝐻 Σ_g (𝑇 ∘ ⟨“⟨𝐴, ∅⟩”⟩)))
33	23, 32	mpdan 674	. 2 ⊢ (𝜑 → (𝐸‘[⟨“⟨𝐴, ∅⟩”⟩] ∼ ) = (𝐻 Σ_g (𝑇 ∘ ⟨“⟨𝐴, ∅⟩”⟩)))
34	24, 25, 26, 27, 1, 28	frgpuptf 18656	. . . . . 6 ⊢ (𝜑 → 𝑇:(𝐼 × 2_o)⟶𝐵)
35		s1co 14057	. . . . . 6 ⊢ ((⟨𝐴, ∅⟩ ∈ (𝐼 × 2_o) ∧ 𝑇:(𝐼 × 2_o)⟶𝐵) → (𝑇 ∘ ⟨“⟨𝐴, ∅⟩”⟩) = ⟨“(𝑇‘⟨𝐴, ∅⟩)”⟩)
36	13, 34, 35	syl2anc 576	. . . . 5 ⊢ (𝜑 → (𝑇 ∘ ⟨“⟨𝐴, ∅⟩”⟩) = ⟨“(𝑇‘⟨𝐴, ∅⟩)”⟩)
37		df-ov 6979	. . . . . . 7 ⊢ (𝐴𝑇∅) = (𝑇‘⟨𝐴, ∅⟩)
38		iftrue 4356	. . . . . . . . . 10 ⊢ (𝑧 = ∅ → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = (𝐹‘𝑦))
39		fveq2 6499	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴))
40	38, 39	sylan9eqr 2837	. . . . . . . . 9 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = ∅) → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = (𝐹‘𝐴))
41		fvex 6512	. . . . . . . . 9 ⊢ (𝐹‘𝐴) ∈ V
42	40, 26, 41	ovmpoa 7121	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2_o) → (𝐴𝑇∅) = (𝐹‘𝐴))
43	2, 11, 42	sylancl 577	. . . . . . 7 ⊢ (𝜑 → (𝐴𝑇∅) = (𝐹‘𝐴))
44	37, 43	syl5eqr 2829	. . . . . 6 ⊢ (𝜑 → (𝑇‘⟨𝐴, ∅⟩) = (𝐹‘𝐴))
45	44	s1eqd 13764	. . . . 5 ⊢ (𝜑 → ⟨“(𝑇‘⟨𝐴, ∅⟩)”⟩ = ⟨“(𝐹‘𝐴)”⟩)
46	36, 45	eqtrd 2815	. . . 4 ⊢ (𝜑 → (𝑇 ∘ ⟨“⟨𝐴, ∅⟩”⟩) = ⟨“(𝐹‘𝐴)”⟩)
47	46	oveq2d 6992	. . 3 ⊢ (𝜑 → (𝐻 Σ_g (𝑇 ∘ ⟨“⟨𝐴, ∅⟩”⟩)) = (𝐻 Σ_g ⟨“(𝐹‘𝐴)”⟩))
48	28, 2	ffvelrnd 6677	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝐵)
49	24	gsumws1 17844	. . . 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 2815	. 2 ⊢ (𝜑 → (𝐻 Σ_g (𝑇 ∘ ⟨“⟨𝐴, ∅⟩”⟩)) = (𝐹‘𝐴))
52	7, 33, 51	3eqtrd 2819	1 ⊢ (𝜑 → (𝐸‘(𝑈‘𝐴)) = (𝐹‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2050 Vcvv 3416 ∅c0 4179 ifcif 4350 {cpr 4443 ⟨cop 4447 ↦ cmpt 5008 I cid 5311 × cxp 5405 ran crn 5408 ∘ ccom 5411 Oncon0 6029 ⟶wf 6184 ‘cfv 6188 (class class class)co 6976 ∈ cmpo 6978 1_oc1o 7898 2_oc2o 7899 [cec 8087 Word cword 13672 ⟨“cs1 13758 Basecbs 16339 Σ_g cgsu 16570 Grpcgrp 17891 inv_gcminusg 17892 ~_FG cefg 18590 freeGrpcfrgp 18591 var_FGrpcvrgp 18592

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-ot 4450 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-oadd 7909 df-er 8089 df-ec 8091 df-qs 8095 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-sup 8701 df-inf 8702 df-card 9162 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-uz 12059 df-fz 12709 df-fzo 12850 df-seq 13185 df-hash 13506 df-word 13673 df-concat 13734 df-s1 13759 df-substr 13804 df-pfx 13853 df-splice 13960 df-s2 14072 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-sca 16437 df-vsca 16438 df-ip 16439 df-tset 16440 df-ple 16441 df-ds 16443 df-0g 16571 df-gsum 16572 df-imas 16637 df-qus 16638 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-submnd 17804 df-frmd 17855 df-grp 17894 df-minusg 17895 df-efg 18593 df-frgp 18594 df-vrgp 18595

This theorem is referenced by: frgpup3 18664

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