Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 18887 | . . . 4 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) → (𝑈‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) |
6 | 1, 2, 5 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝑈‘𝐴) = [〈“〈𝐴, ∅〉”〉] ∼ ) |
7 | 6 | fveq2d 6668 | . 2 ⊢ (𝜑 → (𝐸‘(𝑈‘𝐴)) = (𝐸‘[〈“〈𝐴, ∅〉”〉] ∼ )) |
8 | 0ex 5203 | . . . . . . . 8 ⊢ ∅ ∈ V | |
9 | 8 | prid1 4691 | . . . . . . 7 ⊢ ∅ ∈ {∅, 1o} |
10 | df2o3 8111 | . . . . . . 7 ⊢ 2o = {∅, 1o} | |
11 | 9, 10 | eleqtrri 2912 | . . . . . 6 ⊢ ∅ ∈ 2o |
12 | opelxpi 5586 | . . . . . 6 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2o) → 〈𝐴, ∅〉 ∈ (𝐼 × 2o)) | |
13 | 2, 11, 12 | sylancl 588 | . . . . 5 ⊢ (𝜑 → 〈𝐴, ∅〉 ∈ (𝐼 × 2o)) |
14 | 13 | s1cld 13951 | . . . 4 ⊢ (𝜑 → 〈“〈𝐴, ∅〉”〉 ∈ Word (𝐼 × 2o)) |
15 | frgpup.w | . . . . 5 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
16 | 2on 8105 | . . . . . . 7 ⊢ 2o ∈ On | |
17 | xpexg 7467 | . . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V) | |
18 | 1, 16, 17 | sylancl 588 | . . . . . 6 ⊢ (𝜑 → (𝐼 × 2o) ∈ V) |
19 | wrdexg 13865 | . . . . . 6 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V) | |
20 | fvi 6734 | . . . . . 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 2868 | . . . 4 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2o)) |
23 | 14, 22 | eleqtrrd 2916 | . . 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 18894 | . . 3 ⊢ ((𝜑 ∧ 〈“〈𝐴, ∅〉”〉 ∈ 𝑊) → (𝐸‘[〈“〈𝐴, ∅〉”〉] ∼ ) = (𝐻 Σg (𝑇 ∘ 〈“〈𝐴, ∅〉”〉))) |
33 | 23, 32 | mpdan 685 | . 2 ⊢ (𝜑 → (𝐸‘[〈“〈𝐴, ∅〉”〉] ∼ ) = (𝐻 Σg (𝑇 ∘ 〈“〈𝐴, ∅〉”〉))) |
34 | 24, 25, 26, 27, 1, 28 | frgpuptf 18890 | . . . . . 6 ⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵) |
35 | s1co 14189 | . . . . . 6 ⊢ ((〈𝐴, ∅〉 ∈ (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ 〈“〈𝐴, ∅〉”〉) = 〈“(𝑇‘〈𝐴, ∅〉)”〉) | |
36 | 13, 34, 35 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝑇 ∘ 〈“〈𝐴, ∅〉”〉) = 〈“(𝑇‘〈𝐴, ∅〉)”〉) |
37 | df-ov 7153 | . . . . . . 7 ⊢ (𝐴𝑇∅) = (𝑇‘〈𝐴, ∅〉) | |
38 | iftrue 4472 | . . . . . . . . . 10 ⊢ (𝑧 = ∅ → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = (𝐹‘𝑦)) | |
39 | fveq2 6664 | . . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝐹‘𝑦) = (𝐹‘𝐴)) | |
40 | 38, 39 | sylan9eqr 2878 | . . . . . . . . 9 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = ∅) → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = (𝐹‘𝐴)) |
41 | fvex 6677 | . . . . . . . . 9 ⊢ (𝐹‘𝐴) ∈ V | |
42 | 40, 26, 41 | ovmpoa 7299 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝐼 ∧ ∅ ∈ 2o) → (𝐴𝑇∅) = (𝐹‘𝐴)) |
43 | 2, 11, 42 | sylancl 588 | . . . . . . 7 ⊢ (𝜑 → (𝐴𝑇∅) = (𝐹‘𝐴)) |
44 | 37, 43 | syl5eqr 2870 | . . . . . 6 ⊢ (𝜑 → (𝑇‘〈𝐴, ∅〉) = (𝐹‘𝐴)) |
45 | 44 | s1eqd 13949 | . . . . 5 ⊢ (𝜑 → 〈“(𝑇‘〈𝐴, ∅〉)”〉 = 〈“(𝐹‘𝐴)”〉) |
46 | 36, 45 | eqtrd 2856 | . . . 4 ⊢ (𝜑 → (𝑇 ∘ 〈“〈𝐴, ∅〉”〉) = 〈“(𝐹‘𝐴)”〉) |
47 | 46 | oveq2d 7166 | . . 3 ⊢ (𝜑 → (𝐻 Σg (𝑇 ∘ 〈“〈𝐴, ∅〉”〉)) = (𝐻 Σg 〈“(𝐹‘𝐴)”〉)) |
48 | 28, 2 | ffvelrnd 6846 | . . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝐵) |
49 | 24 | gsumws1 17996 | . . . 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 2856 | . 2 ⊢ (𝜑 → (𝐻 Σg (𝑇 ∘ 〈“〈𝐴, ∅〉”〉)) = (𝐹‘𝐴)) |
52 | 7, 33, 51 | 3eqtrd 2860 | 1 ⊢ (𝜑 → (𝐸‘(𝑈‘𝐴)) = (𝐹‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 ifcif 4466 {cpr 4562 〈cop 4566 ↦ cmpt 5138 I cid 5453 × cxp 5547 ran crn 5550 ∘ ccom 5553 Oncon0 6185 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 1oc1o 8089 2oc2o 8090 [cec 8281 Word cword 13855 〈“cs1 13943 Basecbs 16477 Σg cgsu 16708 Grpcgrp 18097 invgcminusg 18098 ~FG cefg 18826 freeGrpcfrgp 18827 varFGrpcvrgp 18828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-ot 4569 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-ec 8285 df-qs 8289 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 df-substr 13997 df-pfx 14027 df-splice 14106 df-s2 14204 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-0g 16709 df-gsum 16710 df-imas 16775 df-qus 16776 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-frmd 18008 df-grp 18100 df-minusg 18101 df-efg 18829 df-frgp 18830 df-vrgp 18831 |
This theorem is referenced by: frgpup3 18898 |
Copyright terms: Public domain | W3C validator |