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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 | grpmnd 18113 | . . . . . 6 ⊢ (𝐻 ∈ Grp → 𝐻 ∈ Mnd) | |
4 | 2, 3 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
5 | frgpup.w | . . . . . . . 8 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
6 | fviss 6744 | . . . . . . . 8 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o) | |
7 | 5, 6 | eqsstri 4004 | . . . . . . 7 ⊢ 𝑊 ⊆ Word (𝐼 × 2o) |
8 | 7 | sseli 3966 | . . . . . 6 ⊢ (𝑔 ∈ 𝑊 → 𝑔 ∈ Word (𝐼 × 2o)) |
9 | frgpup.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐻) | |
10 | frgpup.n | . . . . . . 7 ⊢ 𝑁 = (invg‘𝐻) | |
11 | frgpup.t | . . . . . . 7 ⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) | |
12 | frgpup.i | . . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
13 | frgpup.a | . . . . . . 7 ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) | |
14 | 9, 10, 11, 2, 12, 13 | frgpuptf 18899 | . . . . . 6 ⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵) |
15 | wrdco 14196 | . . . . . 6 ⊢ ((𝑔 ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ 𝑔) ∈ Word 𝐵) | |
16 | 8, 14, 15 | syl2anr 598 | . . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → (𝑇 ∘ 𝑔) ∈ Word 𝐵) |
17 | 9 | gsumwcl 18006 | . . . . 5 ⊢ ((𝐻 ∈ Mnd ∧ (𝑇 ∘ 𝑔) ∈ Word 𝐵) → (𝐻 Σg (𝑇 ∘ 𝑔)) ∈ 𝐵) |
18 | 4, 16, 17 | syl2an2r 683 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → (𝐻 Σg (𝑇 ∘ 𝑔)) ∈ 𝐵) |
19 | frgpup.r | . . . . . 6 ⊢ ∼ = ( ~FG ‘𝐼) | |
20 | 5, 19 | efger 18847 | . . . . 5 ⊢ ∼ Er 𝑊 |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → ∼ Er 𝑊) |
22 | 5 | fvexi 6687 | . . . . 5 ⊢ 𝑊 ∈ V |
23 | 22 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ V) |
24 | coeq2 5732 | . . . . 5 ⊢ (𝑔 = ℎ → (𝑇 ∘ 𝑔) = (𝑇 ∘ ℎ)) | |
25 | 24 | oveq2d 7175 | . . . 4 ⊢ (𝑔 = ℎ → (𝐻 Σg (𝑇 ∘ 𝑔)) = (𝐻 Σg (𝑇 ∘ ℎ))) |
26 | 9, 10, 11, 2, 12, 13, 5, 19 | frgpuplem 18901 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∼ ℎ) → (𝐻 Σg (𝑇 ∘ 𝑔)) = (𝐻 Σg (𝑇 ∘ ℎ))) |
27 | 1, 18, 21, 23, 25, 26 | qliftfund 8386 | . . 3 ⊢ (𝜑 → Fun 𝐸) |
28 | 1, 18, 21, 23 | qliftf 8388 | . . 3 ⊢ (𝜑 → (Fun 𝐸 ↔ 𝐸:(𝑊 / ∼ )⟶𝐵)) |
29 | 27, 28 | mpbid 234 | . 2 ⊢ (𝜑 → 𝐸:(𝑊 / ∼ )⟶𝐵) |
30 | frgpup.g | . . . . . . 7 ⊢ 𝐺 = (freeGrp‘𝐼) | |
31 | eqid 2824 | . . . . . . 7 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o)) | |
32 | 30, 31, 19 | frgpval 18887 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
33 | 12, 32 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
34 | 2on 8114 | . . . . . . . . 9 ⊢ 2o ∈ On | |
35 | xpexg 7476 | . . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V) | |
36 | 12, 34, 35 | sylancl 588 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 × 2o) ∈ V) |
37 | wrdexg 13874 | . . . . . . . 8 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V) | |
38 | fvi 6743 | . . . . . . . 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 | 5, 39 | syl5eq 2871 | . . . . . 6 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2o)) |
41 | eqid 2824 | . . . . . . . 8 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o))) | |
42 | 31, 41 | frmdbas 18020 | . . . . . . 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 2862 | . . . . 5 ⊢ (𝜑 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2o)))) |
45 | 19 | fvexi 6687 | . . . . . 6 ⊢ ∼ ∈ V |
46 | 45 | a1i 11 | . . . . 5 ⊢ (𝜑 → ∼ ∈ V) |
47 | fvexd 6688 | . . . . 5 ⊢ (𝜑 → (freeMnd‘(𝐼 × 2o)) ∈ V) | |
48 | 33, 44, 46, 47 | qusbas 16821 | . . . 4 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐺)) |
49 | frgpup.x | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
50 | 48, 49 | syl6reqr 2878 | . . 3 ⊢ (𝜑 → 𝑋 = (𝑊 / ∼ )) |
51 | 50 | feq2d 6503 | . 2 ⊢ (𝜑 → (𝐸:𝑋⟶𝐵 ↔ 𝐸:(𝑊 / ∼ )⟶𝐵)) |
52 | 29, 51 | mpbird 259 | 1 ⊢ (𝜑 → 𝐸:𝑋⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∅c0 4294 ifcif 4470 〈cop 4576 ↦ cmpt 5149 I cid 5462 × cxp 5556 ran crn 5559 ∘ ccom 5562 Oncon0 6194 Fun wfun 6352 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 2oc2o 8099 Er wer 8289 [cec 8290 / cqs 8291 Word cword 13864 Basecbs 16486 Σg cgsu 16717 /s cqus 16781 Mndcmnd 17914 freeMndcfrmd 18015 Grpcgrp 18106 invgcminusg 18107 ~FG cefg 18835 freeGrpcfrgp 18836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-ot 4579 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-ec 8294 df-qs 8298 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-inf 8910 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-word 13865 df-concat 13926 df-s1 13953 df-substr 14006 df-pfx 14036 df-splice 14115 df-s2 14213 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-0g 16718 df-gsum 16719 df-imas 16784 df-qus 16785 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-frmd 18017 df-grp 18109 df-minusg 18110 df-efg 18838 df-frgp 18839 |
This theorem is referenced by: frgpupval 18903 frgpup1 18904 |
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