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Mirrors > Home > MPE Home > Th. List > Mathboxes > cycpmco2lem1 | Structured version Visualization version GIF version |
Description: Lemma for cycpmco2 30794. (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 3941 | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐷) |
5 | ssrab2 4049 | . . . . . . . 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 2820 | . . . . . . . . . . . 12 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
9 | 1, 7, 8 | tocycf 30778 | . . . . . . . . . . 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 6516 | . . . . . . . . 9 ⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) |
12 | 6, 11 | eleqtrd 2914 | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}) |
13 | 5, 12 | sseldi 3958 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) |
14 | id 22 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊) | |
15 | dmeq 5765 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊) | |
16 | eqidd 2821 | . . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷) | |
17 | 14, 15, 16 | f1eq123d 6601 | . . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷)) |
18 | 17 | elrab3 3677 | . . . . . . . 8 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ 𝑊:dom 𝑊–1-1→𝐷)) |
19 | 18 | biimpa 479 | . . . . . . 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 586 | . . . . . 6 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) |
21 | f1f 6568 | . . . . . 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 6514 | . . . 4 ⊢ (𝜑 → ran 𝑊 ⊆ 𝐷) |
24 | cycpmco2.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ ran 𝑊) | |
25 | 23, 24 | sseldd 3961 | . . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐷) |
26 | 3 | eldifbd 3942 | . . . . 5 ⊢ (𝜑 → ¬ 𝐼 ∈ ran 𝑊) |
27 | nelne2 3114 | . . . . 5 ⊢ ((𝐽 ∈ ran 𝑊 ∧ ¬ 𝐼 ∈ ran 𝑊) → 𝐽 ≠ 𝐼) | |
28 | 24, 26, 27 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝐽 ≠ 𝐼) |
29 | 28 | necomd 3070 | . . 3 ⊢ (𝜑 → 𝐼 ≠ 𝐽) |
30 | 1, 2, 4, 25, 29, 7 | cyc2fv1 30782 | . 2 ⊢ (𝜑 → ((𝑀‘〈“𝐼𝐽”〉)‘𝐼) = 𝐽) |
31 | 30 | fveq2d 6667 | 1 ⊢ (𝜑 → ((𝑀‘𝑊)‘((𝑀‘〈“𝐼𝐽”〉)‘𝐼)) = ((𝑀‘𝑊)‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2113 ≠ wne 3015 {crab 3141 ∖ cdif 3926 〈cotp 4568 ◡ccnv 5547 dom cdm 5548 ran crn 5549 ⟶wf 6344 –1-1→wf1 6345 ‘cfv 6348 (class class class)co 7149 1c1 10531 + caddc 10533 Word cword 13858 〈“cs1 13942 splice csplice 14104 〈“cs2 14196 Basecbs 16476 SymGrpcsymg 18488 toCycctocyc 30767 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-sup 8899 df-inf 8900 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12890 df-fzo 13031 df-fl 13159 df-mod 13235 df-hash 13688 df-word 13859 df-concat 13916 df-s1 13943 df-substr 13996 df-pfx 14026 df-csh 14144 df-s2 14203 df-struct 16478 df-ndx 16479 df-slot 16480 df-base 16482 df-sets 16483 df-ress 16484 df-plusg 16571 df-tset 16577 df-efmnd 18027 df-symg 18489 df-tocyc 30768 |
This theorem is referenced by: cycpmco2lem4 30790 |
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