cycpmco2lem2 - Mathbox for Thierry Arnoux

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Theorem cycpmco2lem2 33478

Description: Lemma for cycpmco2 33484. (Contributed by Thierry Arnoux, 4-Jan-2024.)

**Hypotheses**
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”⟩⟩)

**Assertion**
Ref	Expression
cycpmco2lem2	⊢ (𝜑 → (𝑈‘𝐸) = 𝐼)

Proof of Theorem cycpmco2lem2 Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cycpmco2.1	. . . . 5 ⊢ 𝑈 = (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩)
2		cycpmco2.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ dom 𝑀)
3		cycpmco2.e	. . . . . . 7 ⊢ 𝐸 = ((^◡𝑊‘𝐽) + 1)
4		ovexd 7458	. . . . . . 7 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ V)
5	3, 4	eqeltrid 2870	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ V)
6		cycpmco2.i	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
7	6	eldifad 3920	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
8	7	s1cld 14662	. . . . . 6 ⊢ (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
9		splval 14812	. . . . . 6 ⊢ ((𝑊 ∈ dom 𝑀 ∧ (𝐸 ∈ V ∧ 𝐸 ∈ V ∧ ⟨“𝐼”⟩ ∈ Word 𝐷)) → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
10	2, 5, 5, 8, 9	syl13anc 1399	. . . . 5 ⊢ (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
11	1, 10	eqtrid 2813	. . . 4 ⊢ (𝜑 → 𝑈 = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
12	11	fveq1d 6890	. . 3 ⊢ (𝜑 → (𝑈‘𝐸) = ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘𝐸))
13		ssrab2 4037	. . . . . . 7 ⊢ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ⊆ Word 𝐷
14		cycpmco2.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
15		cycpmco2.c	. . . . . . . . . . 11 ⊢ 𝑀 = (toCyc‘𝐷)
16		cycpmco2.s	. . . . . . . . . . 11 ⊢ 𝑆 = (SymGrp‘𝐷)
17		eqid 2766	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
18	15, 16, 17	tocycf 33468	. . . . . . . . . 10 ⊢ (𝐷 ∈ 𝑉 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆))
19	14, 18	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆))
20	19	fdmd 6723	. . . . . . . 8 ⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
21	2, 20	eleqtrd 2868	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
22	13, 21	sselid 3938	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
23		pfxcl 14739	. . . . . 6 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
24	22, 23	syl 18	. . . . 5 ⊢ (𝜑 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
25		ccatcl 14631	. . . . 5 ⊢ (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
26	24, 8, 25	syl2anc 596	. . . 4 ⊢ (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
27		swrdcl 14705	. . . . 5 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
28	22, 27	syl 18	. . . 4 ⊢ (𝜑 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
29		fz0ssnn0 13669	. . . . . . 7 ⊢ (0...(♯‘𝑊)) ⊆ ℕ₀
30		id 23	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
31		dmeq 5898	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
32		eqidd 2767	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷)
33	30, 31, 32	f1eq123d 6819	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷))
34	33	elrab 3653	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
35	21, 34	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
36	35	simprd 501	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
37		f1cnv 6852	. . . . . . . . . . . 12 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → ^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊)
38		f1of 6827	. . . . . . . . . . . 12 ⊢ (^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊 → ^◡𝑊:ran 𝑊⟶dom 𝑊)
39	36, 37, 38	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → ^◡𝑊:ran 𝑊⟶dom 𝑊)
40		cycpmco2.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ ran 𝑊)
41	39, 40	ffvelcdmd 7087	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ dom 𝑊)
42		wrddm 14578	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
43	22, 42	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
44	41, 43	eleqtrd 2868	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)))
45		fzofzp1 13812	. . . . . . . . 9 ⊢ ((^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)) → ((^◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊)))
46	44, 45	syl 18	. . . . . . . 8 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊)))
47	3, 46	eqeltrid 2870	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (0...(♯‘𝑊)))
48	29, 47	sselid 3938	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
49		fzonn0p1 13790	. . . . . 6 ⊢ (𝐸 ∈ ℕ₀ → 𝐸 ∈ (0..^(𝐸 + 1)))
50	48, 49	syl 18	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (0..^(𝐸 + 1)))
51		ccatws1len 14680	. . . . . . . 8 ⊢ ((𝑊 prefix 𝐸) ∈ Word 𝐷 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = ((♯‘(𝑊 prefix 𝐸)) + 1))
52	22, 23, 51	3syl 19	. . . . . . 7 ⊢ (𝜑 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = ((♯‘(𝑊 prefix 𝐸)) + 1))
53		pfxlen 14745	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
54	22, 47, 53	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
55	54	oveq1d 7438	. . . . . . 7 ⊢ (𝜑 → ((♯‘(𝑊 prefix 𝐸)) + 1) = (𝐸 + 1))
56	52, 55	eqtrd 2801	. . . . . 6 ⊢ (𝜑 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = (𝐸 + 1))
57	56	oveq2d 7439	. . . . 5 ⊢ (𝜑 → (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))) = (0..^(𝐸 + 1)))
58	50, 57	eleqtrrd 2869	. . . 4 ⊢ (𝜑 → 𝐸 ∈ (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))))
59		ccatval1 14634	. . . 4 ⊢ ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷 ∧ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷 ∧ 𝐸 ∈ (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)))) → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘𝐸) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘𝐸))
60	26, 28, 58, 59	syl3anc 1398	. . 3 ⊢ (𝜑 → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘𝐸) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘𝐸))
61	48	nn0zd 12634	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℤ)
62		elfzomin 13785	. . . . . 6 ⊢ (𝐸 ∈ ℤ → 𝐸 ∈ (𝐸..^(𝐸 + 1)))
63	61, 62	syl 18	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (𝐸..^(𝐸 + 1)))
64		s1len 14665	. . . . . . . 8 ⊢ (♯‘⟨“𝐼”⟩) = 1
65	64	a1i 11	. . . . . . 7 ⊢ (𝜑 → (♯‘⟨“𝐼”⟩) = 1)
66	54, 65	oveq12d 7441	. . . . . 6 ⊢ (𝜑 → ((♯‘(𝑊 prefix 𝐸)) + (♯‘⟨“𝐼”⟩)) = (𝐸 + 1))
67	54, 66	oveq12d 7441	. . . . 5 ⊢ (𝜑 → ((♯‘(𝑊 prefix 𝐸))..^((♯‘(𝑊 prefix 𝐸)) + (♯‘⟨“𝐼”⟩))) = (𝐸..^(𝐸 + 1)))
68	63, 67	eleqtrrd 2869	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ((♯‘(𝑊 prefix 𝐸))..^((♯‘(𝑊 prefix 𝐸)) + (♯‘⟨“𝐼”⟩))))
69		ccatval2 14635	. . . 4 ⊢ (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷 ∧ 𝐸 ∈ ((♯‘(𝑊 prefix 𝐸))..^((♯‘(𝑊 prefix 𝐸)) + (♯‘⟨“𝐼”⟩)))) → (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘𝐸) = (⟨“𝐼”⟩‘(𝐸 − (♯‘(𝑊 prefix 𝐸)))))
70	24, 8, 68, 69	syl3anc 1398	. . 3 ⊢ (𝜑 → (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘𝐸) = (⟨“𝐼”⟩‘(𝐸 − (♯‘(𝑊 prefix 𝐸)))))
71	12, 60, 70	3eqtrd 2805	. 2 ⊢ (𝜑 → (𝑈‘𝐸) = (⟨“𝐼”⟩‘(𝐸 − (♯‘(𝑊 prefix 𝐸)))))
72	54	oveq2d 7439	. . . 4 ⊢ (𝜑 → (𝐸 − (♯‘(𝑊 prefix 𝐸))) = (𝐸 − 𝐸))
73	48	nn0cnd 12585	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℂ)
74	73	subidd 11575	. . . 4 ⊢ (𝜑 → (𝐸 − 𝐸) = 0)
75	72, 74	eqtrd 2801	. . 3 ⊢ (𝜑 → (𝐸 − (♯‘(𝑊 prefix 𝐸))) = 0)
76	75	fveq2d 6892	. 2 ⊢ (𝜑 → (⟨“𝐼”⟩‘(𝐸 − (♯‘(𝑊 prefix 𝐸)))) = (⟨“𝐼”⟩‘0))
77		s1fv 14670	. . 3 ⊢ (𝐼 ∈ (𝐷 ∖ ran 𝑊) → (⟨“𝐼”⟩‘0) = 𝐼)
78	6, 77	syl 18	. 2 ⊢ (𝜑 → (⟨“𝐼”⟩‘0) = 𝐼)
79	71, 76, 78	3eqtrd 2805	1 ⊢ (𝜑 → (𝑈‘𝐸) = 𝐼)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 {crab 3419 Vcvv 3458 ∖ cdif 3905 ⟨cop 4600 ⟨cotp 4602 ^◡ccnv 5665 dom cdm 5666 ran crn 5667 ⟶wf 6539 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7423 0cc0 11118 1c1 11119 + caddc 11121 − cmin 11459 ℕ₀cn0 12522 ℤcz 12609 ...cfz 13553 ..^cfzo 13701 ♯chash 14386 Word cword 14570 ++ cconcat 14627 ⟨“cs1 14654 substr csubstr 14700 prefix cpfx 14732 splice csplice 14810 Basecbs 17294 SymGrpcsymg 19470 toCycctocyc 33457

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-ot 4603 df-uni 4878 df-int 4918 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-1st 7995 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8703 df-map 8835 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-inf 9413 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-uz 12881 df-rp 13035 df-fz 13554 df-fzo 13702 df-fl 13845 df-mod 13923 df-hash 14387 df-word 14571 df-concat 14628 df-s1 14655 df-substr 14701 df-pfx 14733 df-splice 14811 df-csh 14852 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-tset 17354 df-efmnd 18959 df-symg 19471 df-tocyc 33458

This theorem is used by: cycpmco2lem4 33480 cycpmco2lem5 33481 cycpmco2lem6 33482 cycpmco2 33484

W3C validator