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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 7205 | . 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 18962 | . . 3 ⊢ ∼ Er 𝑊 |
6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → ∼ Er 𝑊) |
7 | 3 | fvexi 6688 | . . 3 ⊢ 𝑊 ∈ V |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → 𝑊 ∈ V) |
9 | coeq2 5701 | . . 3 ⊢ (𝑔 = 𝐴 → (𝑇 ∘ 𝑔) = (𝑇 ∘ 𝐴)) | |
10 | 9 | oveq2d 7186 | . 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 19017 | . . 3 ⊢ (𝜑 → 𝐸:𝑋⟶𝐵) |
20 | 19 | ffund 6508 | . 2 ⊢ (𝜑 → Fun 𝐸) |
21 | 1, 2, 6, 8, 10, 20 | qliftval 8417 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑊) → (𝐸‘[𝐴] ∼ ) = (𝐻 Σg (𝑇 ∘ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ∅c0 4211 ifcif 4414 〈cop 4522 ↦ cmpt 5110 I cid 5428 × cxp 5523 ran crn 5526 ∘ ccom 5529 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 ∈ cmpo 7172 2oc2o 8125 Er wer 8317 [cec 8318 Word cword 13955 Basecbs 16586 Σg cgsu 16817 Grpcgrp 18219 invgcminusg 18220 ~FG cefg 18950 freeGrpcfrgp 18951 |
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 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
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 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-ot 4525 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-2o 8132 df-er 8320 df-ec 8322 df-qs 8326 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-sup 8979 df-inf 8980 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-fz 12982 df-fzo 13125 df-seq 13461 df-hash 13783 df-word 13956 df-concat 14012 df-s1 14039 df-substr 14092 df-pfx 14122 df-splice 14201 df-s2 14299 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-0g 16818 df-gsum 16819 df-imas 16884 df-qus 16885 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-submnd 18073 df-frmd 18130 df-grp 18222 df-minusg 18223 df-efg 18953 df-frgp 18954 |
This theorem is referenced by: frgpup1 19019 frgpup2 19020 frgpup3lem 19021 |
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