Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > cycpmco2lem1 | Structured version Visualization version GIF version |
Description: Lemma for cycpmco2 31119. (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 3878 | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐷) |
5 | ssrab2 3993 | . . . . . . . 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 2737 | . . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
9 | 1, 7, 8 | tocycf 31103 | . . . . . . . . . . 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 6556 | . . . . . . . . 9 ⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) |
12 | 6, 11 | eleqtrd 2840 | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) |
13 | 5, 12 | sseldi 3899 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) |
14 | id 22 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊) | |
15 | dmeq 5772 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊) | |
16 | eqidd 2738 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷) | |
17 | 14, 15, 16 | f1eq123d 6653 | . . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷)) |
18 | 17 | elrab3 3603 | . . . . . . . 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 6615 | . . . . . 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 6553 | . . . 4 ⊢ (𝜑 → ran 𝑊 ⊆ 𝐷) |
24 | cycpmco2.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) | |
25 | 23, 24 | sseldd 3902 | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐷) |
26 | 3 | eldifbd 3879 | . . . . 5 ⊢ (𝜑 → ¬ 𝐼 ∈ ran 𝑊) |
27 | nelne2 3039 | . . . . 5 ⊢ ((𝐽 ∈ ran 𝑊 ∧ ¬ 𝐼 ∈ ran 𝑊) → 𝐽 ≠ 𝐼) | |
28 | 24, 26, 27 | syl2anc 587 | . . . 4 ⊢ (𝜑 → 𝐽 ≠ 𝐼) |
29 | 28 | necomd 2996 | . . 3 ⊢ (𝜑 → 𝐼 ≠ 𝐽) |
30 | 1, 2, 4, 25, 29, 7 | cyc2fv1 31107 | . 2 ⊢ (𝜑 → ((𝑀‘〈“𝐼𝐽”〉)‘𝐼) = 𝐽) |
31 | 30 | fveq2d 6721 | 1 ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘〈“𝐼𝐽”〉)‘𝐼)) = ((𝑀‘𝑊)‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1543 ∈ wcel 2110 ≠ wne 2940 {crab 3065 ∖ cdif 3863 〈cotp 4549 ◡ccnv 5550 dom cdm 5551 ran crn 5552 ⟶wf 6376 –1-1→wf1 6377 ‘cfv 6380 (class class class)co 7213 1c1 10730 + caddc 10732 Word cword 14069 〈“cs1 14152 splice csplice 14314 〈“cs2 14406 Basecbs 16760 SymGrpcsymg 18759 toCycctocyc 31092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-inf 9059 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-hash 13897 df-word 14070 df-concat 14126 df-s1 14153 df-substr 14206 df-pfx 14236 df-csh 14354 df-s2 14413 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-tset 16821 df-efmnd 18296 df-symg 18760 df-tocyc 31093 |
This theorem is referenced by: cycpmco2lem4 31115 |
Copyright terms: Public domain | W3C validator |