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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 18589 | . . . . 5 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
4 | frgpup.w | . . . . . . . 8 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
5 | fviss 6845 | . . . . . . . 8 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o) | |
6 | 4, 5 | eqsstri 3955 | . . . . . . 7 ⊢ 𝑊 ⊆ Word (𝐼 × 2o) |
7 | 6 | sseli 3917 | . . . . . 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 19376 | . . . . . 6 ⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵) |
14 | wrdco 14544 | . . . . . 6 ⊢ ((𝑔 ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ 𝑔) ∈ Word 𝐵) | |
15 | 7, 13, 14 | syl2anr 597 | . . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → (𝑇 ∘ 𝑔) ∈ Word 𝐵) |
16 | 8 | gsumwcl 18477 | . . . . 5 ⊢ ((𝐻 ∈ Mnd ∧ (𝑇 ∘ 𝑔) ∈ Word 𝐵) → (𝐻 Σg (𝑇 ∘ 𝑔)) ∈ 𝐵) |
17 | 3, 15, 16 | syl2an2r 682 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → (𝐻 Σg (𝑇 ∘ 𝑔)) ∈ 𝐵) |
18 | frgpup.r | . . . . . 6 ⊢ ∼ = ( ~FG ‘𝐼) | |
19 | 4, 18 | efger 19324 | . . . . 5 ⊢ ∼ Er 𝑊 |
20 | 19 | a1i 11 | . . . 4 ⊢ (𝜑 → ∼ Er 𝑊) |
21 | 4 | fvexi 6788 | . . . . 5 ⊢ 𝑊 ∈ V |
22 | 21 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ V) |
23 | coeq2 5767 | . . . . 5 ⊢ (𝑔 = ℎ → (𝑇 ∘ 𝑔) = (𝑇 ∘ ℎ)) | |
24 | 23 | oveq2d 7291 | . . . 4 ⊢ (𝑔 = ℎ → (𝐻 Σg (𝑇 ∘ 𝑔)) = (𝐻 Σg (𝑇 ∘ ℎ))) |
25 | 8, 9, 10, 2, 11, 12, 4, 18 | frgpuplem 19378 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∼ ℎ) → (𝐻 Σg (𝑇 ∘ 𝑔)) = (𝐻 Σg (𝑇 ∘ ℎ))) |
26 | 1, 17, 20, 22, 24, 25 | qliftfund 8592 | . . 3 ⊢ (𝜑 → Fun 𝐸) |
27 | 1, 17, 20, 22 | qliftf 8594 | . . 3 ⊢ (𝜑 → (Fun 𝐸 ↔ 𝐸:(𝑊 / ∼ )⟶𝐵)) |
28 | 26, 27 | mpbid 231 | . 2 ⊢ (𝜑 → 𝐸:(𝑊 / ∼ )⟶𝐵) |
29 | frgpup.x | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
30 | frgpup.g | . . . . . . 7 ⊢ 𝐺 = (freeGrp‘𝐼) | |
31 | eqid 2738 | . . . . . . 7 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o)) | |
32 | 30, 31, 18 | frgpval 19364 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
33 | 11, 32 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
34 | 2on 8311 | . . . . . . . . 9 ⊢ 2o ∈ On | |
35 | xpexg 7600 | . . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V) | |
36 | 11, 34, 35 | sylancl 586 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 × 2o) ∈ V) |
37 | wrdexg 14227 | . . . . . . . 8 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V) | |
38 | fvi 6844 | . . . . . . . 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 | eqtrid 2790 | . . . . . 6 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2o)) |
41 | eqid 2738 | . . . . . . . 8 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o))) | |
42 | 31, 41 | frmdbas 18491 | . . . . . . 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 2781 | . . . . 5 ⊢ (𝜑 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2o)))) |
45 | 18 | fvexi 6788 | . . . . . 6 ⊢ ∼ ∈ V |
46 | 45 | a1i 11 | . . . . 5 ⊢ (𝜑 → ∼ ∈ V) |
47 | fvexd 6789 | . . . . 5 ⊢ (𝜑 → (freeMnd‘(𝐼 × 2o)) ∈ V) | |
48 | 33, 44, 46, 47 | qusbas 17256 | . . . 4 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐺)) |
49 | 29, 48 | eqtr4id 2797 | . . 3 ⊢ (𝜑 → 𝑋 = (𝑊 / ∼ )) |
50 | 49 | feq2d 6586 | . 2 ⊢ (𝜑 → (𝐸:𝑋⟶𝐵 ↔ 𝐸:(𝑊 / ∼ )⟶𝐵)) |
51 | 28, 50 | mpbird 256 | 1 ⊢ (𝜑 → 𝐸:𝑋⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ∅c0 4256 ifcif 4459 〈cop 4567 ↦ cmpt 5157 I cid 5488 × cxp 5587 ran crn 5590 ∘ ccom 5593 Oncon0 6266 Fun wfun 6427 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 2oc2o 8291 Er wer 8495 [cec 8496 / cqs 8497 Word cword 14217 Basecbs 16912 Σg cgsu 17151 /s cqus 17216 Mndcmnd 18385 freeMndcfrmd 18486 Grpcgrp 18577 invgcminusg 18578 ~FG cefg 19312 freeGrpcfrgp 19313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-ec 8500 df-qs 8504 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-word 14218 df-concat 14274 df-s1 14301 df-substr 14354 df-pfx 14384 df-splice 14463 df-s2 14561 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-0g 17152 df-gsum 17153 df-imas 17219 df-qus 17220 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-frmd 18488 df-grp 18580 df-minusg 18581 df-efg 19315 df-frgp 19316 |
This theorem is referenced by: frgpupval 19380 frgpup1 19381 |
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