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Mirrors > Home > MPE Home > Th. List > frmdup3 | Structured version Visualization version GIF version |
Description: Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
frmdup3.m | ⊢ 𝑀 = (freeMnd‘𝐼) |
frmdup3.b | ⊢ 𝐵 = (Base‘𝐺) |
frmdup3.u | ⊢ 𝑈 = (varFMnd‘𝐼) |
Ref | Expression |
---|---|
frmdup3 | ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → ∃!𝑚 ∈ (𝑀 MndHom 𝐺)(𝑚 ∘ 𝑈) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frmdup3.m | . . 3 ⊢ 𝑀 = (freeMnd‘𝐼) | |
2 | frmdup3.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | eqid 2821 | . . 3 ⊢ (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) | |
4 | simp1 1132 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → 𝐺 ∈ Mnd) | |
5 | simp2 1133 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → 𝐼 ∈ 𝑉) | |
6 | simp3 1134 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → 𝐴:𝐼⟶𝐵) | |
7 | 1, 2, 3, 4, 5, 6 | frmdup1 18023 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) ∈ (𝑀 MndHom 𝐺)) |
8 | 4 | adantr 483 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) ∧ 𝑦 ∈ 𝐼) → 𝐺 ∈ Mnd) |
9 | 5 | adantr 483 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ 𝑉) |
10 | 6 | adantr 483 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) ∧ 𝑦 ∈ 𝐼) → 𝐴:𝐼⟶𝐵) |
11 | frmdup3.u | . . . . 5 ⊢ 𝑈 = (varFMnd‘𝐼) | |
12 | simpr 487 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼) | |
13 | 1, 2, 3, 8, 9, 10, 11, 12 | frmdup2 18024 | . . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) ∧ 𝑦 ∈ 𝐼) → ((𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥)))‘(𝑈‘𝑦)) = (𝐴‘𝑦)) |
14 | 13 | mpteq2dva 5154 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → (𝑦 ∈ 𝐼 ↦ ((𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥)))‘(𝑈‘𝑦))) = (𝑦 ∈ 𝐼 ↦ (𝐴‘𝑦))) |
15 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
16 | 15, 2 | mhmf 17955 | . . . . 5 ⊢ ((𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) ∈ (𝑀 MndHom 𝐺) → (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))):(Base‘𝑀)⟶𝐵) |
17 | 7, 16 | syl 17 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))):(Base‘𝑀)⟶𝐵) |
18 | 11 | vrmdf 18017 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 𝑈:𝐼⟶Word 𝐼) |
19 | 18 | 3ad2ant2 1130 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → 𝑈:𝐼⟶Word 𝐼) |
20 | 1, 15 | frmdbas 18011 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑀) = Word 𝐼) |
21 | 20 | 3ad2ant2 1130 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → (Base‘𝑀) = Word 𝐼) |
22 | 21 | feq3d 6496 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → (𝑈:𝐼⟶(Base‘𝑀) ↔ 𝑈:𝐼⟶Word 𝐼)) |
23 | 19, 22 | mpbird 259 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → 𝑈:𝐼⟶(Base‘𝑀)) |
24 | fcompt 6890 | . . . 4 ⊢ (((𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))):(Base‘𝑀)⟶𝐵 ∧ 𝑈:𝐼⟶(Base‘𝑀)) → ((𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) ∘ 𝑈) = (𝑦 ∈ 𝐼 ↦ ((𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥)))‘(𝑈‘𝑦)))) | |
25 | 17, 23, 24 | syl2anc 586 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → ((𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) ∘ 𝑈) = (𝑦 ∈ 𝐼 ↦ ((𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥)))‘(𝑈‘𝑦)))) |
26 | 6 | feqmptd 6728 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → 𝐴 = (𝑦 ∈ 𝐼 ↦ (𝐴‘𝑦))) |
27 | 14, 25, 26 | 3eqtr4d 2866 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → ((𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) ∘ 𝑈) = 𝐴) |
28 | 1, 2, 11 | frmdup3lem 18025 | . . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) ∧ (𝑚 ∈ (𝑀 MndHom 𝐺) ∧ (𝑚 ∘ 𝑈) = 𝐴)) → 𝑚 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥)))) |
29 | 28 | expr 459 | . . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) ∧ 𝑚 ∈ (𝑀 MndHom 𝐺)) → ((𝑚 ∘ 𝑈) = 𝐴 → 𝑚 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))))) |
30 | 29 | ralrimiva 3182 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → ∀𝑚 ∈ (𝑀 MndHom 𝐺)((𝑚 ∘ 𝑈) = 𝐴 → 𝑚 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))))) |
31 | coeq1 5723 | . . . 4 ⊢ (𝑚 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) → (𝑚 ∘ 𝑈) = ((𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) ∘ 𝑈)) | |
32 | 31 | eqeq1d 2823 | . . 3 ⊢ (𝑚 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) → ((𝑚 ∘ 𝑈) = 𝐴 ↔ ((𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) ∘ 𝑈) = 𝐴)) |
33 | 32 | eqreu 3720 | . 2 ⊢ (((𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) ∈ (𝑀 MndHom 𝐺) ∧ ((𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) ∘ 𝑈) = 𝐴 ∧ ∀𝑚 ∈ (𝑀 MndHom 𝐺)((𝑚 ∘ 𝑈) = 𝐴 → 𝑚 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))))) → ∃!𝑚 ∈ (𝑀 MndHom 𝐺)(𝑚 ∘ 𝑈) = 𝐴) |
34 | 7, 27, 30, 33 | syl3anc 1367 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → ∃!𝑚 ∈ (𝑀 MndHom 𝐺)(𝑚 ∘ 𝑈) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃!wreu 3140 ↦ cmpt 5139 ∘ ccom 5554 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 Word cword 13855 Basecbs 16477 Σg cgsu 16708 Mndcmnd 17905 MndHom cmhm 17948 freeMndcfrmd 18006 varFMndcvrmd 18007 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-word 13856 df-lsw 13909 df-concat 13917 df-s1 13944 df-substr 13997 df-pfx 14027 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-0g 16709 df-gsum 16710 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-frmd 18008 df-vrmd 18009 |
This theorem is referenced by: (None) |
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