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Mirrors > Home > MPE Home > Th. List > frgpupf | Structured version Visualization version GIF version |
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.) |
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))〉) |
Ref | Expression |
---|---|
frgpupf | ⊢ (𝜑 → 𝐸:𝑋⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgpup.e | . . . 4 ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ 〈[𝑔] ∼ , (𝐻 Σg (𝑇 ∘ 𝑔))〉) | |
2 | frgpup.h | . . . . . 6 ⊢ (𝜑 → 𝐻 ∈ Grp) | |
3 | grpmnd 18048 | . . . . . 6 ⊢ (𝐻 ∈ Grp → 𝐻 ∈ Mnd) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
5 | frgpup.w | . . . . . . . 8 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
6 | fviss 6734 | . . . . . . . 8 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o) | |
7 | 5, 6 | eqsstri 3998 | . . . . . . 7 ⊢ 𝑊 ⊆ Word (𝐼 × 2o) |
8 | 7 | sseli 3960 | . . . . . 6 ⊢ (𝑔 ∈ 𝑊 → 𝑔 ∈ Word (𝐼 × 2o)) |
9 | frgpup.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐻) | |
10 | frgpup.n | . . . . . . 7 ⊢ 𝑁 = (invg‘𝐻) | |
11 | frgpup.t | . . . . . . 7 ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) | |
12 | frgpup.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
13 | frgpup.a | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) | |
14 | 9, 10, 11, 2, 12, 13 | frgpuptf 18825 | . . . . . 6 ⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵) |
15 | wrdco 14181 | . . . . . 6 ⊢ ((𝑔 ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ 𝑔) ∈ Word 𝐵) | |
16 | 8, 14, 15 | syl2anr 596 | . . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → (𝑇 ∘ 𝑔) ∈ Word 𝐵) |
17 | 9 | gsumwcl 17991 | . . . . 5 ⊢ ((𝐻 ∈ Mnd ∧ (𝑇 ∘ 𝑔) ∈ Word 𝐵) → (𝐻 Σg (𝑇 ∘ 𝑔)) ∈ 𝐵) |
18 | 4, 16, 17 | syl2an2r 681 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → (𝐻 Σg (𝑇 ∘ 𝑔)) ∈ 𝐵) |
19 | frgpup.r | . . . . . 6 ⊢ ∼ = ( ~FG ‘𝐼) | |
20 | 5, 19 | efger 18773 | . . . . 5 ⊢ ∼ Er 𝑊 |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → ∼ Er 𝑊) |
22 | 5 | fvexi 6677 | . . . . 5 ⊢ 𝑊 ∈ V |
23 | 22 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ V) |
24 | coeq2 5722 | . . . . 5 ⊢ (𝑔 = ℎ → (𝑇 ∘ 𝑔) = (𝑇 ∘ ℎ)) | |
25 | 24 | oveq2d 7161 | . . . 4 ⊢ (𝑔 = ℎ → (𝐻 Σg (𝑇 ∘ 𝑔)) = (𝐻 Σg (𝑇 ∘ ℎ))) |
26 | 9, 10, 11, 2, 12, 13, 5, 19 | frgpuplem 18827 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∼ ℎ) → (𝐻 Σg (𝑇 ∘ 𝑔)) = (𝐻 Σg (𝑇 ∘ ℎ))) |
27 | 1, 18, 21, 23, 25, 26 | qliftfund 8372 | . . 3 ⊢ (𝜑 → Fun 𝐸) |
28 | 1, 18, 21, 23 | qliftf 8374 | . . 3 ⊢ (𝜑 → (Fun 𝐸 ↔ 𝐸:(𝑊 / ∼ )⟶𝐵)) |
29 | 27, 28 | mpbid 233 | . 2 ⊢ (𝜑 → 𝐸:(𝑊 / ∼ )⟶𝐵) |
30 | frgpup.g | . . . . . . 7 ⊢ 𝐺 = (freeGrp‘𝐼) | |
31 | eqid 2818 | . . . . . . 7 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o)) | |
32 | 30, 31, 19 | frgpval 18813 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
33 | 12, 32 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
34 | 2on 8100 | . . . . . . . . 9 ⊢ 2o ∈ On | |
35 | xpexg 7462 | . . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V) | |
36 | 12, 34, 35 | sylancl 586 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 × 2o) ∈ V) |
37 | wrdexg 13859 | . . . . . . . 8 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V) | |
38 | fvi 6733 | . . . . . . . 8 ⊢ (Word (𝐼 × 2o) ∈ V → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o)) | |
39 | 36, 37, 38 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 × 2o)) |
40 | 5, 39 | syl5eq 2865 | . . . . . 6 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2o)) |
41 | eqid 2818 | . . . . . . . 8 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o))) | |
42 | 31, 41 | frmdbas 18005 | . . . . . . 7 ⊢ ((𝐼 × 2o) ∈ V → (Base‘(freeMnd‘(𝐼 × 2o))) = Word (𝐼 × 2o)) |
43 | 36, 42 | syl 17 | . . . . . 6 ⊢ (𝜑 → (Base‘(freeMnd‘(𝐼 × 2o))) = Word (𝐼 × 2o)) |
44 | 40, 43 | eqtr4d 2856 | . . . . 5 ⊢ (𝜑 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2o)))) |
45 | 19 | fvexi 6677 | . . . . . 6 ⊢ ∼ ∈ V |
46 | 45 | a1i 11 | . . . . 5 ⊢ (𝜑 → ∼ ∈ V) |
47 | fvexd 6678 | . . . . 5 ⊢ (𝜑 → (freeMnd‘(𝐼 × 2o)) ∈ V) | |
48 | 33, 44, 46, 47 | qusbas 16806 | . . . 4 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐺)) |
49 | frgpup.x | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
50 | 48, 49 | syl6reqr 2872 | . . 3 ⊢ (𝜑 → 𝑋 = (𝑊 / ∼ )) |
51 | 50 | feq2d 6493 | . 2 ⊢ (𝜑 → (𝐸:𝑋⟶𝐵 ↔ 𝐸:(𝑊 / ∼ )⟶𝐵)) |
52 | 29, 51 | mpbird 258 | 1 ⊢ (𝜑 → 𝐸:𝑋⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 ifcif 4463 〈cop 4563 ↦ cmpt 5137 I cid 5452 × cxp 5546 ran crn 5549 ∘ ccom 5552 Oncon0 6184 Fun wfun 6342 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ∈ cmpo 7147 2oc2o 8085 Er wer 8275 [cec 8276 / cqs 8277 Word cword 13849 Basecbs 16471 Σg cgsu 16702 /s cqus 16766 Mndcmnd 17899 freeMndcfrmd 18000 Grpcgrp 18041 invgcminusg 18042 ~FG cefg 18761 freeGrpcfrgp 18762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-ot 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-ec 8280 df-qs 8284 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-word 13850 df-concat 13911 df-s1 13938 df-substr 13991 df-pfx 14021 df-splice 14100 df-s2 14198 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-0g 16703 df-gsum 16704 df-imas 16769 df-qus 16770 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-frmd 18002 df-grp 18044 df-minusg 18045 df-efg 18764 df-frgp 18765 |
This theorem is referenced by: frgpupval 18829 frgpup1 18830 |
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