| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 18964 | . . . . 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 3979 | . . . . . 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 19788 | . . . . . 6 ⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵) |
| 14 | wrdco 14870 | . . . . . 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 18852 | . . . . 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 19736 | . . . . 5 ⊢ ∼ Er 𝑊 |
| 20 | 19 | a1i 11 | . . . 4 ⊢ (𝜑 → ∼ Er 𝑊) |
| 21 | 4 | fvexi 6920 | . . . . 5 ⊢ 𝑊 ∈ V |
| 22 | 21 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ V) |
| 23 | coeq2 5869 | . . . . 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 19790 | . . . 4 ⊢ ((𝜑 ∧ 𝑔 ∼ ℎ) → (𝐻 Σg (𝑇 ∘ 𝑔)) = (𝐻 Σg (𝑇 ∘ ℎ))) |
| 26 | 1, 17, 20, 22, 24, 25 | qliftfund 8843 | . . 3 ⊢ (𝜑 → Fun 𝐸) |
| 27 | 1, 17, 20, 22 | qliftf 8845 | . . 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 2737 | . . . . . . 7 ⊢ (freeMnd‘(𝐼 × 2o)) = (freeMnd‘(𝐼 × 2o)) | |
| 32 | 30, 31, 18 | frgpval 19776 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
| 33 | 11, 32 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐺 = ((freeMnd‘(𝐼 × 2o)) /s ∼ )) |
| 34 | 2on 8520 | . . . . . . . . 9 ⊢ 2o ∈ On | |
| 35 | xpexg 7770 | . . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) → (𝐼 × 2o) ∈ V) | |
| 36 | 11, 34, 35 | sylancl 586 | . . . . . . . 8 ⊢ (𝜑 → (𝐼 × 2o) ∈ V) |
| 37 | wrdexg 14562 | . . . . . . . 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 2789 | . . . . . 6 ⊢ (𝜑 → 𝑊 = Word (𝐼 × 2o)) |
| 41 | eqid 2737 | . . . . . . . 8 ⊢ (Base‘(freeMnd‘(𝐼 × 2o))) = (Base‘(freeMnd‘(𝐼 × 2o))) | |
| 42 | 31, 41 | frmdbas 18865 | . . . . . . 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 2780 | . . . . 5 ⊢ (𝜑 → 𝑊 = (Base‘(freeMnd‘(𝐼 × 2o)))) |
| 45 | 18 | fvexi 6920 | . . . . . 6 ⊢ ∼ ∈ V |
| 46 | 45 | a1i 11 | . . . . 5 ⊢ (𝜑 → ∼ ∈ V) |
| 47 | fvexd 6921 | . . . . 5 ⊢ (𝜑 → (freeMnd‘(𝐼 × 2o)) ∈ V) | |
| 48 | 33, 44, 46, 47 | qusbas 17590 | . . . 4 ⊢ (𝜑 → (𝑊 / ∼ ) = (Base‘𝐺)) |
| 49 | 29, 48 | eqtr4id 2796 | . . 3 ⊢ (𝜑 → 𝑋 = (𝑊 / ∼ )) |
| 50 | 49 | feq2d 6722 | . 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 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ifcif 4525 〈cop 4632 ↦ cmpt 5225 I cid 5577 × cxp 5683 ran crn 5686 ∘ ccom 5689 Oncon0 6384 Fun wfun 6555 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 2oc2o 8500 Er wer 8742 [cec 8743 / cqs 8744 Word cword 14552 Basecbs 17247 Σg cgsu 17485 /s cqus 17550 Mndcmnd 18747 freeMndcfrmd 18860 Grpcgrp 18951 invgcminusg 18952 ~FG cefg 19724 freeGrpcfrgp 19725 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-ec 8747 df-qs 8751 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-substr 14679 df-pfx 14709 df-splice 14788 df-s2 14887 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-0g 17486 df-gsum 17487 df-imas 17553 df-qus 17554 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-frmd 18862 df-grp 18954 df-minusg 18955 df-efg 19727 df-frgp 19728 |
| This theorem is referenced by: frgpupval 19792 frgpup1 19793 |
| Copyright terms: Public domain | W3C validator |