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Mirrors > Home > MPE Home > Th. List > frgpupval | 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 |
---|---|
frgpupval | ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑊) → (𝐸‘[𝐴] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgpup.e | . 2 ⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ 〈[𝑔] ∼ , (𝐻 Σg (𝑇 ∘ 𝑔))〉) | |
2 | ovexd 7185 | . 2 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → (𝐻 Σg (𝑇 ∘ 𝑔)) ∈ V) | |
3 | frgpup.w | . . . 4 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
4 | frgpup.r | . . . 4 ⊢ ∼ = ( ~FG ‘𝐼) | |
5 | 3, 4 | efger 18838 | . . 3 ⊢ ∼ Er 𝑊 |
6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → ∼ Er 𝑊) |
7 | 3 | fvexi 6678 | . . 3 ⊢ 𝑊 ∈ V |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → 𝑊 ∈ V) |
9 | coeq2 5723 | . . 3 ⊢ (𝑔 = 𝐴 → (𝑇 ∘ 𝑔) = (𝑇 ∘ 𝐴)) | |
10 | 9 | oveq2d 7166 | . 2 ⊢ (𝑔 = 𝐴 → (𝐻 Σg (𝑇 ∘ 𝑔)) = (𝐻 Σg (𝑇 ∘ 𝐴))) |
11 | frgpup.b | . . . 4 ⊢ 𝐵 = (Base‘𝐻) | |
12 | frgpup.n | . . . 4 ⊢ 𝑁 = (invg‘𝐻) | |
13 | frgpup.t | . . . 4 ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) | |
14 | frgpup.h | . . . 4 ⊢ (𝜑 → 𝐻 ∈ Grp) | |
15 | frgpup.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
16 | frgpup.a | . . . 4 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) | |
17 | frgpup.g | . . . 4 ⊢ 𝐺 = (freeGrp‘𝐼) | |
18 | frgpup.x | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
19 | 11, 12, 13, 14, 15, 16, 3, 4, 17, 18, 1 | frgpupf 18893 | . . 3 ⊢ (𝜑 → 𝐸:𝑋⟶𝐵) |
20 | 19 | ffund 6512 | . 2 ⊢ (𝜑 → Fun 𝐸) |
21 | 1, 2, 6, 8, 10, 20 | qliftval 8380 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑊) → (𝐸‘[𝐴] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 ifcif 4466 〈cop 4566 ↦ cmpt 5138 I cid 5453 × cxp 5547 ran crn 5550 ∘ ccom 5553 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ∈ cmpo 7152 2oc2o 8090 Er wer 8280 [cec 8281 Word cword 13855 Basecbs 16477 Σg cgsu 16708 Grpcgrp 18097 invgcminusg 18098 ~FG cefg 18826 freeGrpcfrgp 18827 |
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 |
This theorem is referenced by: frgpup1 18895 frgpup2 18896 frgpup3lem 18897 |
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