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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 2610 | . . 3 ⊢ (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) | |
4 | simp1 1054 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → 𝐺 ∈ Mnd) | |
5 | simp2 1055 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → 𝐼 ∈ 𝑉) | |
6 | simp3 1056 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → 𝐴:𝐼⟶𝐵) | |
7 | 1, 2, 3, 4, 5, 6 | frmdup1 17224 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) ∈ (𝑀 MndHom 𝐺)) |
8 | 4 | adantr 480 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) ∧ 𝑦 ∈ 𝐼) → 𝐺 ∈ Mnd) |
9 | 5 | adantr 480 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) ∧ 𝑦 ∈ 𝐼) → 𝐼 ∈ 𝑉) |
10 | 6 | adantr 480 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) ∧ 𝑦 ∈ 𝐼) → 𝐴:𝐼⟶𝐵) |
11 | frmdup3.u | . . . . 5 ⊢ 𝑈 = (varFMnd‘𝐼) | |
12 | simpr 476 | . . . . 5 ⊢ (((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) ∧ 𝑦 ∈ 𝐼) → 𝑦 ∈ 𝐼) | |
13 | 1, 2, 3, 8, 9, 10, 11, 12 | frmdup2 17225 | . . . 4 ⊢ (((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) ∧ 𝑦 ∈ 𝐼) → ((𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥)))‘(𝑈‘𝑦)) = (𝐴‘𝑦)) |
14 | 13 | mpteq2dva 4672 | . . 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 2610 | . . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
16 | 15, 2 | mhmf 17163 | . . . . 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 17218 | . . . . . 6 ⊢ (𝐼 ∈ 𝑉 → 𝑈:𝐼⟶Word 𝐼) |
19 | 18 | 3ad2ant2 1076 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → 𝑈:𝐼⟶Word 𝐼) |
20 | 1, 15 | frmdbas 17212 | . . . . . . 7 ⊢ (𝐼 ∈ 𝑉 → (Base‘𝑀) = Word 𝐼) |
21 | 20 | 3ad2ant2 1076 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → (Base‘𝑀) = Word 𝐼) |
22 | 21 | feq3d 5945 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → (𝑈:𝐼⟶(Base‘𝑀) ↔ 𝑈:𝐼⟶Word 𝐼)) |
23 | 19, 22 | mpbird 246 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → 𝑈:𝐼⟶(Base‘𝑀)) |
24 | fcompt 6306 | . . . 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 691 | . . 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 6159 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → 𝐴 = (𝑦 ∈ 𝐼 ↦ (𝐴‘𝑦))) |
27 | 14, 25, 26 | 3eqtr4d 2654 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝐼 ∈ 𝑉 ∧ 𝐴:𝐼⟶𝐵) → ((𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) ∘ 𝑈) = 𝐴) |
28 | 1, 2, 11 | frmdup3lem 17226 | . . . 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 641 | . . 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 2949 | . 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 5201 | . . . 4 ⊢ (𝑚 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) → (𝑚 ∘ 𝑈) = ((𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) ∘ 𝑈)) | |
32 | 31 | eqeq1d 2612 | . . 3 ⊢ (𝑚 = (𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) → ((𝑚 ∘ 𝑈) = 𝐴 ↔ ((𝑥 ∈ Word 𝐼 ↦ (𝐺 Σg (𝐴 ∘ 𝑥))) ∘ 𝑈) = 𝐴)) |
33 | 32 | eqreu 3365 | . 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 1318 | 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 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃!wreu 2898 ↦ cmpt 4643 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Word cword 13146 Basecbs 15695 Σg cgsu 15924 Mndcmnd 17117 MndHom cmhm 17156 freeMndcfrmd 17207 varFMndcvrmd 17208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-frmd 17209 df-vrmd 17210 |
This theorem is referenced by: (None) |
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