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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 18977 | . . . . 5 ⊢ (𝜑 → 𝐻 ∈ Mnd) |
4 | frgpup.w | . . . . . . . 8 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
5 | fviss 6986 | . . . . . . . 8 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o) | |
6 | 4, 5 | eqsstri 4030 | . . . . . . 7 ⊢ 𝑊 ⊆ Word (𝐼 × 2o) |
7 | 6 | sseli 3991 | . . . . . 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 19803 | . . . . . 6 ⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵) |
14 | wrdco 14867 | . . . . . 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 18865 | . . . . 5 ⊢ ((𝐻 ∈ Mnd ∧ (𝑇 ∘ 𝑔) ∈ Word 𝐵) → (𝐻 Σg (𝑇 ∘ 𝑔)) ∈ 𝐵) |
17 | 3, 15, 16 | syl2an2r 685 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑊) → (𝐻 Σg (𝑇 ∘ 𝑔)) ∈ 𝐵) |
18 | frgpup.r | . . . . . 6 ⊢ ∼ = ( ~FG ‘𝐼) | |
19 | 4, 18 | efger 19751 | . . . . 5 ⊢ ∼ Er 𝑊 |
20 | 19 | a1i 11 | . . . 4 ⊢ (𝜑 → ∼ Er 𝑊) |
21 | 4 | fvexi 6921 | . . . . 5 ⊢ 𝑊 ∈ V |
22 | 21 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ V) |
23 | coeq2 5872 | . . . . 5 ⊢ (𝑔 = ℎ → (𝑇 ∘ 𝑔) = (𝑇 ∘ ℎ)) | |
24 | 23 | oveq2d 7447 | . . . 4 ⊢ (𝑔 = ℎ → (𝐻 Σg (𝑇 ∘ 𝑔)) = (𝐻 Σg (𝑇 ∘ ℎ))) |
25 | 8, 9, 10, 2, 11, 12, 4, 18 | frgpuplem 19805 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∼ ℎ) → (𝐻 Σg (𝑇 ∘ 𝑔)) = (𝐻 Σg (𝑇 ∘ ℎ))) |
26 | 1, 17, 20, 22, 24, 25 | qliftfund 8842 | . . 3 ⊢ (𝜑 → Fun 𝐸) |
27 | 1, 17, 20, 22 | qliftf 8844 | . . 3 ⊢ (𝜑 → (Fun 𝐸 ↔ 𝐸:(𝑊 / ∼ )⟶𝐵)) |
28 | 26, 27 | mpbid 232 | . 2 ⊢ (𝜑 → 𝐸:(𝑊 / ∼ )⟶𝐵) |
29 | frgpup.x | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
30 | frgpup.g | . . . . . . 7 ⊢ 𝐺 = (freeGrp‘𝐼) | |
31 | eqid 2735 | . . . . . . 7 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o)) | |
32 | 30, 31, 18 | frgpval 19791 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
33 | 11, 32 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
34 | 2on 8519 | . . . . . . . . 9 ⊢ 2o ∈ On | |
35 | xpexg 7769 | . . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V) | |
36 | 11, 34, 35 | sylancl 586 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 × 2o) ∈ V) |
37 | wrdexg 14559 | . . . . . . . 8 ⊢ ((𝐼 × 2o) ∈ V → Word (𝐼 × 2o) ∈ V) | |
38 | fvi 6985 | . . . . . . . 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 2787 | . . . . . 6 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2o)) |
41 | eqid 2735 | . . . . . . . 8 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o))) | |
42 | 31, 41 | frmdbas 18878 | . . . . . . 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 2778 | . . . . 5 ⊢ (𝜑 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2o)))) |
45 | 18 | fvexi 6921 | . . . . . 6 ⊢ ∼ ∈ V |
46 | 45 | a1i 11 | . . . . 5 ⊢ (𝜑 → ∼ ∈ V) |
47 | fvexd 6922 | . . . . 5 ⊢ (𝜑 → (freeMnd‘(𝐼 × 2o)) ∈ V) | |
48 | 33, 44, 46, 47 | qusbas 17592 | . . . 4 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐺)) |
49 | 29, 48 | eqtr4id 2794 | . . 3 ⊢ (𝜑 → 𝑋 = (𝑊 / ∼ )) |
50 | 49 | feq2d 6723 | . 2 ⊢ (𝜑 → (𝐸:𝑋⟶𝐵 ↔ 𝐸:(𝑊 / ∼ )⟶𝐵)) |
51 | 28, 50 | mpbird 257 | 1 ⊢ (𝜑 → 𝐸:𝑋⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 ifcif 4531 〈cop 4637 ↦ cmpt 5231 I cid 5582 × cxp 5687 ran crn 5690 ∘ ccom 5693 Oncon0 6386 Fun wfun 6557 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 2oc2o 8499 Er wer 8741 [cec 8742 / cqs 8743 Word cword 14549 Basecbs 17245 Σg cgsu 17487 /s cqus 17552 Mndcmnd 18760 freeMndcfrmd 18873 Grpcgrp 18964 invgcminusg 18965 ~FG cefg 19739 freeGrpcfrgp 19740 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-substr 14676 df-pfx 14706 df-splice 14785 df-s2 14884 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-0g 17488 df-gsum 17489 df-imas 17555 df-qus 17556 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-frmd 18875 df-grp 18967 df-minusg 18968 df-efg 19742 df-frgp 19743 |
This theorem is referenced by: frgpupval 19807 frgpup1 19808 |
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