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Mirrors > Home > MPE Home > Th. List > frgpup2 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
frgpup.b | ⊢ 𝐵 = (Base‘𝐻) |
frgpup.n | ⊢ 𝑁 = (invg‘𝐻) |
frgpup.t | ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ 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 × 2𝑜)) |
frgpup.r | ⊢ ∼ = ( ~FG ‘𝐼) |
frgpup.g | ⊢ 𝐺 = (freeGrp‘𝐼) |
frgpup.x | ⊢ 𝑋 = (Base‘𝐺) |
frgpup.e | ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ 〈[𝑔] ∼ , (𝐻 Σg (𝑇 ∘ 𝑔))〉) |
frgpup.u | ⊢ 𝑈 = (varFGrp‘𝐼) |
frgpup.y | ⊢ (𝜑 → 𝐴 ∈ 𝐼) |
Ref | Expression |
---|---|
frgpup2 | ⊢ (𝜑 → (𝐸‘(𝑈‘𝐴)) = (𝐹‘𝐴)) |
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 = (varFGrp‘𝐼) | |
5 | 3, 4 | vrgpval 18003 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) |
6 | 1, 2, 5 | syl2anc 691 | . . 3 ⊢ (𝜑 → (𝑈‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) |
7 | 6 | fveq2d 6107 | . 2 ⊢ (𝜑 → (𝐸‘(𝑈‘𝐴)) = (𝐸‘[〈“〈𝐴, ∅〉”〉] ∼ )) |
8 | 0ex 4718 | . . . . . . . 8 ⊢ ∅ ∈ V | |
9 | 8 | prid1 4241 | . . . . . . 7 ⊢ ∅ ∈ {∅, 1𝑜} |
10 | df2o3 7460 | . . . . . . 7 ⊢ 2𝑜 = {∅, 1𝑜} | |
11 | 9, 10 | eleqtrri 2687 | . . . . . 6 ⊢ ∅ ∈ 2𝑜 |
12 | opelxpi 5072 | . . . . . 6 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2𝑜) → 〈𝐴, ∅〉 ∈ (𝐼 × 2𝑜)) | |
13 | 2, 11, 12 | sylancl 693 | . . . . 5 ⊢ (𝜑 → 〈𝐴, ∅〉 ∈ (𝐼 × 2𝑜)) |
14 | 13 | s1cld 13236 | . . . 4 ⊢ (𝜑 → 〈“〈𝐴, ∅〉”〉 ∈ Word (𝐼 × 2𝑜)) |
15 | frgpup.w | . . . . 5 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜)) | |
16 | 2on 7455 | . . . . . . 7 ⊢ 2𝑜 ∈ On | |
17 | xpexg 6858 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈ V) | |
18 | 1, 16, 17 | sylancl 693 | . . . . . 6 ⊢ (𝜑 → (𝐼 × 2𝑜) ∈ V) |
19 | wrdexg 13170 | . . . . . 6 ⊢ ((𝐼 × 2𝑜) ∈ V → Word (𝐼 × 2𝑜) ∈ V) | |
20 | fvi 6165 | . . . . . 6 ⊢ (Word (𝐼 × 2𝑜) ∈ V → ( I ‘Word (𝐼 × 2𝑜)) = Word (𝐼 × 2𝑜)) | |
21 | 18, 19, 20 | 3syl 18 | . . . . 5 ⊢ (𝜑 → ( I ‘Word (𝐼 × 2𝑜)) = Word (𝐼 × 2𝑜)) |
22 | 15, 21 | syl5eq 2656 | . . . 4 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2𝑜)) |
23 | 14, 22 | eleqtrrd 2691 | . . 3 ⊢ (𝜑 → 〈“〈𝐴, ∅〉”〉 ∈ 𝑊) |
24 | frgpup.b | . . . 4 ⊢ 𝐵 = (Base‘𝐻) | |
25 | frgpup.n | . . . 4 ⊢ 𝑁 = (invg‘𝐻) | |
26 | frgpup.t | . . . 4 ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ 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 Σg (𝑇 ∘ 𝑔))〉) | |
32 | 24, 25, 26, 27, 1, 28, 15, 3, 29, 30, 31 | frgpupval 18010 | . . 3 ⊢ ((𝜑 ∧ 〈“〈𝐴, ∅〉”〉 ∈ 𝑊) → (𝐸‘[〈“〈𝐴, ∅〉”〉] ∼ ) = (𝐻 Σg (𝑇 ∘ 〈“〈𝐴, ∅〉”〉))) |
33 | 23, 32 | mpdan 699 | . 2 ⊢ (𝜑 → (𝐸‘[〈“〈𝐴, ∅〉”〉] ∼ ) = (𝐻 Σg (𝑇 ∘ 〈“〈𝐴, ∅〉”〉))) |
34 | 24, 25, 26, 27, 1, 28 | frgpuptf 18006 | . . . . . 6 ⊢ (𝜑 → 𝑇:(𝐼 × 2𝑜)⟶𝐵) |
35 | s1co 13430 | . . . . . 6 ⊢ ((〈𝐴, ∅〉 ∈ (𝐼 × 2𝑜) ∧ 𝑇:(𝐼 × 2𝑜)⟶𝐵) → (𝑇 ∘ 〈“〈𝐴, ∅〉”〉) = 〈“(𝑇‘〈𝐴, ∅〉)”〉) | |
36 | 13, 34, 35 | syl2anc 691 | . . . . 5 ⊢ (𝜑 → (𝑇 ∘ 〈“〈𝐴, ∅〉”〉) = 〈“(𝑇‘〈𝐴, ∅〉)”〉) |
37 | df-ov 6552 | . . . . . . 7 ⊢ (𝐴𝑇∅) = (𝑇‘〈𝐴, ∅〉) | |
38 | iftrue 4042 | . . . . . . . . . 10 ⊢ (𝑧 = ∅ → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = (𝐹‘𝑦)) | |
39 | fveq2 6103 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) | |
40 | 38, 39 | sylan9eqr 2666 | . . . . . . . . 9 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = ∅) → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = (𝐹‘𝐴)) |
41 | fvex 6113 | . . . . . . . . 9 ⊢ (𝐹‘𝐴) ∈ V | |
42 | 40, 26, 41 | ovmpt2a 6689 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2𝑜) → (𝐴𝑇∅) = (𝐹‘𝐴)) |
43 | 2, 11, 42 | sylancl 693 | . . . . . . 7 ⊢ (𝜑 → (𝐴𝑇∅) = (𝐹‘𝐴)) |
44 | 37, 43 | syl5eqr 2658 | . . . . . 6 ⊢ (𝜑 → (𝑇‘〈𝐴, ∅〉) = (𝐹‘𝐴)) |
45 | 44 | s1eqd 13234 | . . . . 5 ⊢ (𝜑 → 〈“(𝑇‘〈𝐴, ∅〉)”〉 = 〈“(𝐹‘𝐴)”〉) |
46 | 36, 45 | eqtrd 2644 | . . . 4 ⊢ (𝜑 → (𝑇 ∘ 〈“〈𝐴, ∅〉”〉) = 〈“(𝐹‘𝐴)”〉) |
47 | 46 | oveq2d 6565 | . . 3 ⊢ (𝜑 → (𝐻 Σg (𝑇 ∘ 〈“〈𝐴, ∅〉”〉)) = (𝐻 Σg 〈“(𝐹‘𝐴)”〉)) |
48 | 28, 2 | ffvelrnd 6268 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝐵) |
49 | 24 | gsumws1 17199 | . . . 4 ⊢ ((𝐹‘𝐴) ∈ 𝐵 → (𝐻 Σg 〈“(𝐹‘𝐴)”〉) = (𝐹‘𝐴)) |
50 | 48, 49 | syl 17 | . . 3 ⊢ (𝜑 → (𝐻 Σg 〈“(𝐹‘𝐴)”〉) = (𝐹‘𝐴)) |
51 | 47, 50 | eqtrd 2644 | . 2 ⊢ (𝜑 → (𝐻 Σg (𝑇 ∘ 〈“〈𝐴, ∅〉”〉)) = (𝐹‘𝐴)) |
52 | 7, 33, 51 | 3eqtrd 2648 | 1 ⊢ (𝜑 → (𝐸‘(𝑈‘𝐴)) = (𝐹‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 ifcif 4036 {cpr 4127 〈cop 4131 ↦ cmpt 4643 I cid 4948 × cxp 5036 ran crn 5039 ∘ ccom 5042 Oncon0 5640 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1𝑜c1o 7440 2𝑜c2o 7441 [cec 7627 Word cword 13146 〈“cs1 13149 Basecbs 15695 Σg cgsu 15924 Grpcgrp 17245 invgcminusg 17246 ~FG cefg 17942 freeGrpcfrgp 17943 varFGrpcvrgp 17944 |
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 df-vrgp 17947 |
This theorem is referenced by: frgpup3 18014 |
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