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Mirrors > Home > MPE Home > Th. List > Mathboxes > cycpmco2lem1 | Structured version Visualization version GIF version |
Description: Lemma for cycpmco2 30825. (Contributed by Thierry Arnoux, 4-Jan-2024.) |
Ref | Expression |
---|---|
cycpmco2.c | ⊢ 𝑀 = (toCyc‘𝐷) |
cycpmco2.s | ⊢ 𝑆 = (SymGrp‘𝐷) |
cycpmco2.d | ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
cycpmco2.w | ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) |
cycpmco2.i | ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) |
cycpmco2.j | ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) |
cycpmco2.e | ⊢ 𝐸 = ((◡𝑊‘𝐽) + 1) |
cycpmco2.1 | ⊢ 𝑈 = (𝑊 splice 〈𝐸, 𝐸, 〈“𝐼”〉〉) |
Ref | Expression |
---|---|
cycpmco2lem1 | ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘〈“𝐼𝐽”〉)‘𝐼)) = ((𝑀‘𝑊)‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cycpmco2.c | . . 3 ⊢ 𝑀 = (toCyc‘𝐷) | |
2 | cycpmco2.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
3 | cycpmco2.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊)) | |
4 | 3 | eldifad 3893 | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐷) |
5 | ssrab2 4007 | . . . . . . . 8 ⊢ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ⊆ Word 𝐷 | |
6 | cycpmco2.w | . . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ dom 𝑀) | |
7 | cycpmco2.s | . . . . . . . . . . . 12 ⊢ 𝑆 = (SymGrp‘𝐷) | |
8 | eqid 2798 | . . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
9 | 1, 7, 8 | tocycf 30809 | . . . . . . . . . . 11 ⊢ (𝐷 ∈ 𝑉 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆)) |
10 | 2, 9 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆)) |
11 | 10 | fdmd 6497 | . . . . . . . . 9 ⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) |
12 | 6, 11 | eleqtrd 2892 | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) |
13 | 5, 12 | sseldi 3913 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) |
14 | id 22 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊) | |
15 | dmeq 5736 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊) | |
16 | eqidd 2799 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷) | |
17 | 14, 15, 16 | f1eq123d 6583 | . . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷)) |
18 | 17 | elrab3 3629 | . . . . . . . 8 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ 𝑊:dom 𝑊–1-1→𝐷)) |
19 | 18 | biimpa 480 | . . . . . . 7 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) → 𝑊:dom 𝑊–1-1→𝐷) |
20 | 13, 12, 19 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) |
21 | f1f 6549 | . . . . . 6 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → 𝑊:dom 𝑊⟶𝐷) | |
22 | 20, 21 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊:dom 𝑊⟶𝐷) |
23 | 22 | frnd 6494 | . . . 4 ⊢ (𝜑 → ran 𝑊 ⊆ 𝐷) |
24 | cycpmco2.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) | |
25 | 23, 24 | sseldd 3916 | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐷) |
26 | 3 | eldifbd 3894 | . . . . 5 ⊢ (𝜑 → ¬ 𝐼 ∈ ran 𝑊) |
27 | nelne2 3084 | . . . . 5 ⊢ ((𝐽 ∈ ran 𝑊 ∧ ¬ 𝐼 ∈ ran 𝑊) → 𝐽 ≠ 𝐼) | |
28 | 24, 26, 27 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 𝐽 ≠ 𝐼) |
29 | 28 | necomd 3042 | . . 3 ⊢ (𝜑 → 𝐼 ≠ 𝐽) |
30 | 1, 2, 4, 25, 29, 7 | cyc2fv1 30813 | . 2 ⊢ (𝜑 → ((𝑀‘〈“𝐼𝐽”〉)‘𝐼) = 𝐽) |
31 | 30 | fveq2d 6649 | 1 ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘〈“𝐼𝐽”〉)‘𝐼)) = ((𝑀‘𝑊)‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 {crab 3110 ∖ cdif 3878 〈cotp 4533 ◡ccnv 5518 dom cdm 5519 ran crn 5520 ⟶wf 6320 –1-1→wf1 6321 ‘cfv 6324 (class class class)co 7135 1c1 10527 + caddc 10529 Word cword 13857 〈“cs1 13940 splice csplice 14102 〈“cs2 14194 Basecbs 16475 SymGrpcsymg 18487 toCycctocyc 30798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-fl 13157 df-mod 13233 df-hash 13687 df-word 13858 df-concat 13914 df-s1 13941 df-substr 13994 df-pfx 14024 df-csh 14142 df-s2 14201 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-tset 16576 df-efmnd 18026 df-symg 18488 df-tocyc 30799 |
This theorem is referenced by: cycpmco2lem4 30821 |
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