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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 6956 | . 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 18515 | . . 3 ⊢ ∼ Er 𝑊 |
6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → ∼ Er 𝑊) |
7 | 3 | fvexi 6460 | . . 3 ⊢ 𝑊 ∈ V |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → 𝑊 ∈ V) |
9 | coeq2 5526 | . . 3 ⊢ (𝑔 = 𝐴 → (𝑇 ∘ 𝑔) = (𝑇 ∘ 𝐴)) | |
10 | 9 | oveq2d 6938 | . 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 18572 | . . 3 ⊢ (𝜑 → 𝐸:𝑋⟶𝐵) |
20 | 19 | ffund 6295 | . 2 ⊢ (𝜑 → Fun 𝐸) |
21 | 1, 2, 6, 8, 10, 20 | qliftval 8119 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑊) → (𝐸‘[𝐴] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ∅c0 4141 ifcif 4307 〈cop 4404 ↦ cmpt 4965 I cid 5260 × cxp 5353 ran crn 5356 ∘ ccom 5359 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ↦ cmpt2 6924 2oc2o 7837 Er wer 8023 [cec 8024 Word cword 13599 Basecbs 16255 Σg cgsu 16487 Grpcgrp 17809 invgcminusg 17810 ~FG cefg 18503 freeGrpcfrgp 18504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-ot 4407 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-ec 8028 df-qs 8032 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-inf 8637 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-fzo 12785 df-seq 13120 df-hash 13436 df-word 13600 df-concat 13661 df-s1 13686 df-substr 13731 df-pfx 13780 df-splice 13887 df-s2 13999 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-0g 16488 df-gsum 16489 df-imas 16554 df-qus 16555 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-frmd 17773 df-grp 17812 df-minusg 17813 df-efg 18506 df-frgp 18507 |
This theorem is referenced by: frgpup1 18574 frgpup2 18575 frgpup3lem 18576 |
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