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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 | 2 | grpmndd 18234 | . . . . 5 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
4 | frgpup.w | . . . . . . . 8 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
5 | fviss 6748 | . . . . . . . 8 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o) | |
6 | 4, 5 | eqsstri 3912 | . . . . . . 7 ⊢ 𝑊 ⊆ Word (𝐼 × 2o) |
7 | 6 | sseli 3874 | . . . . . 6 ⊢ (𝑔 ∈ 𝑊 → 𝑔 ∈ Word (𝐼 × 2o)) |
8 | frgpup.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐻) | |
9 | frgpup.n | . . . . . . 7 ⊢ 𝑁 = (invg‘𝐻) | |
10 | frgpup.t | . . . . . . 7 ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) | |
11 | frgpup.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
12 | frgpup.a | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) | |
13 | 8, 9, 10, 2, 11, 12 | frgpuptf 19017 | . . . . . 6 ⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵) |
14 | wrdco 14285 | . . . . . 6 ⊢ ((𝑔 ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ 𝑔) ∈ Word 𝐵) | |
15 | 7, 13, 14 | syl2anr 600 | . . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → (𝑇 ∘ 𝑔) ∈ Word 𝐵) |
16 | 8 | gsumwcl 18122 | . . . . 5 ⊢ ((𝐻 ∈ Mnd ∧ (𝑇 ∘ 𝑔) ∈ Word 𝐵) → (𝐻 Σg (𝑇 ∘ 𝑔)) ∈ 𝐵) |
17 | 3, 15, 16 | syl2an2r 685 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → (𝐻 Σg (𝑇 ∘ 𝑔)) ∈ 𝐵) |
18 | frgpup.r | . . . . . 6 ⊢ ∼ = ( ~FG ‘𝐼) | |
19 | 4, 18 | efger 18965 | . . . . 5 ⊢ ∼ Er 𝑊 |
20 | 19 | a1i 11 | . . . 4 ⊢ (𝜑 → ∼ Er 𝑊) |
21 | 4 | fvexi 6691 | . . . . 5 ⊢ 𝑊 ∈ V |
22 | 21 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ V) |
23 | coeq2 5702 | . . . . 5 ⊢ (𝑔 = ℎ → (𝑇 ∘ 𝑔) = (𝑇 ∘ ℎ)) | |
24 | 23 | oveq2d 7189 | . . . 4 ⊢ (𝑔 = ℎ → (𝐻 Σg (𝑇 ∘ 𝑔)) = (𝐻 Σg (𝑇 ∘ ℎ))) |
25 | 8, 9, 10, 2, 11, 12, 4, 18 | frgpuplem 19019 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∼ ℎ) → (𝐻 Σg (𝑇 ∘ 𝑔)) = (𝐻 Σg (𝑇 ∘ ℎ))) |
26 | 1, 17, 20, 22, 24, 25 | qliftfund 8417 | . . 3 ⊢ (𝜑 → Fun 𝐸) |
27 | 1, 17, 20, 22 | qliftf 8419 | . . 3 ⊢ (𝜑 → (Fun 𝐸 ↔ 𝐸:(𝑊 / ∼ )⟶𝐵)) |
28 | 26, 27 | mpbid 235 | . 2 ⊢ (𝜑 → 𝐸:(𝑊 / ∼ )⟶𝐵) |
29 | frgpup.x | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
30 | frgpup.g | . . . . . . 7 ⊢ 𝐺 = (freeGrp‘𝐼) | |
31 | eqid 2739 | . . . . . . 7 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o)) | |
32 | 30, 31, 18 | frgpval 19005 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
33 | 11, 32 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
34 | 2on 8142 | . . . . . . . . 9 ⊢ 2o ∈ On | |
35 | xpexg 7494 | . . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V) | |
36 | 11, 34, 35 | sylancl 589 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 × 2o) ∈ V) |
37 | wrdexg 13968 | . . . . . . . 8 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V) | |
38 | fvi 6747 | . . . . . . . 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 | 4, 39 | syl5eq 2786 | . . . . . 6 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2o)) |
41 | eqid 2739 | . . . . . . . 8 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o))) | |
42 | 31, 41 | frmdbas 18136 | . . . . . . 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 2777 | . . . . 5 ⊢ (𝜑 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2o)))) |
45 | 18 | fvexi 6691 | . . . . . 6 ⊢ ∼ ∈ V |
46 | 45 | a1i 11 | . . . . 5 ⊢ (𝜑 → ∼ ∈ V) |
47 | fvexd 6692 | . . . . 5 ⊢ (𝜑 → (freeMnd‘(𝐼 × 2o)) ∈ V) | |
48 | 33, 44, 46, 47 | qusbas 16924 | . . . 4 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐺)) |
49 | 29, 48 | eqtr4id 2793 | . . 3 ⊢ (𝜑 → 𝑋 = (𝑊 / ∼ )) |
50 | 49 | feq2d 6491 | . 2 ⊢ (𝜑 → (𝐸:𝑋⟶𝐵 ↔ 𝐸:(𝑊 / ∼ )⟶𝐵)) |
51 | 28, 50 | mpbird 260 | 1 ⊢ (𝜑 → 𝐸:𝑋⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3399 ∅c0 4212 ifcif 4415 〈cop 4523 ↦ cmpt 5111 I cid 5429 × cxp 5524 ran crn 5527 ∘ ccom 5530 Oncon0 6173 Fun wfun 6334 ⟶wf 6336 ‘cfv 6340 (class class class)co 7173 ∈ cmpo 7175 2oc2o 8128 Er wer 8320 [cec 8321 / cqs 8322 Word cword 13958 Basecbs 16589 Σg cgsu 16820 /s cqus 16884 Mndcmnd 18030 freeMndcfrmd 18131 Grpcgrp 18222 invgcminusg 18223 ~FG cefg 18953 freeGrpcfrgp 18954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-ot 4526 df-uni 4798 df-int 4838 df-iun 4884 df-iin 4885 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-1st 7717 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-2o 8135 df-er 8323 df-ec 8325 df-qs 8329 df-map 8442 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-sup 8982 df-inf 8983 df-card 9444 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-7 11787 df-8 11788 df-9 11789 df-n0 11980 df-z 12066 df-dec 12183 df-uz 12328 df-fz 12985 df-fzo 13128 df-seq 13464 df-hash 13786 df-word 13959 df-concat 14015 df-s1 14042 df-substr 14095 df-pfx 14125 df-splice 14204 df-s2 14302 df-struct 16591 df-ndx 16592 df-slot 16593 df-base 16595 df-sets 16596 df-ress 16597 df-plusg 16684 df-mulr 16685 df-sca 16687 df-vsca 16688 df-ip 16689 df-tset 16690 df-ple 16691 df-ds 16693 df-0g 16821 df-gsum 16822 df-imas 16887 df-qus 16888 df-mgm 17971 df-sgrp 18020 df-mnd 18031 df-submnd 18076 df-frmd 18133 df-grp 18225 df-minusg 18226 df-efg 18956 df-frgp 18957 |
This theorem is referenced by: frgpupval 19021 frgpup1 19022 |
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