cycpmco2lem2 - Mathbox for Thierry Arnoux

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Theorem cycpmco2lem2 30769

Description: Lemma for cycpmco2 30775. (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 7191	. . . . . . 7 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ V)
5	3, 4	eqeltrid 2917	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ V)
6		cycpmco2.i	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
7	6	eldifad 3948	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
8	7	s1cld 13957	. . . . . 6 ⊢ (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
9		splval 14113	. . . . . 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 1368	. . . . 5 ⊢ (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
11	1, 10	syl5eq 2868	. . . 4 ⊢ (𝜑 → 𝑈 = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
12	11	fveq1d 6672	. . 3 ⊢ (𝜑 → (𝑈‘𝐸) = ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘𝐸))
13		ssrab2 4056	. . . . . . 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 2821	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
18	15, 16, 17	tocycf 30759	. . . . . . . . . 10 ⊢ (𝐷 ∈ 𝑉 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆))
19	14, 18	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀:{𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷}⟶(Base‘𝑆))
20	19	fdmd 6523	. . . . . . . 8 ⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
21	2, 20	eleqtrd 2915	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
22	13, 21	sseldi 3965	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
23		pfxcl 14039	. . . . . 6 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
24	22, 23	syl 17	. . . . 5 ⊢ (𝜑 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
25		ccatcl 13926	. . . . 5 ⊢ (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
26	24, 8, 25	syl2anc 586	. . . 4 ⊢ (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
27		swrdcl 14007	. . . . 5 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
28	22, 27	syl 17	. . . 4 ⊢ (𝜑 → (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩) ∈ Word 𝐷)
29		fz0ssnn0 13003	. . . . . . 7 ⊢ (0...(♯‘𝑊)) ⊆ ℕ₀
30		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
31		dmeq 5772	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
32		eqidd 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷)
33	30, 31, 32	f1eq123d 6608	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷))
34	33	elrab 3680	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
35	21, 34	sylib 220	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
36	35	simprd 498	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
37		f1cnv 6638	. . . . . . . . . . . 12 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → ^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊)
38		f1of 6615	. . . . . . . . . . . 12 ⊢ (^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊 → ^◡𝑊:ran 𝑊⟶dom 𝑊)
39	36, 37, 38	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ^◡𝑊:ran 𝑊⟶dom 𝑊)
40		cycpmco2.j	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ ran 𝑊)
41	39, 40	ffvelrnd 6852	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ dom 𝑊)
42		wrddm 13869	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
43	22, 42	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
44	41, 43	eleqtrd 2915	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)))
45		fzofzp1 13135	. . . . . . . . 9 ⊢ ((^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)) → ((^◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊)))
46	44, 45	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ (0...(♯‘𝑊)))
47	3, 46	eqeltrid 2917	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (0...(♯‘𝑊)))
48	29, 47	sseldi 3965	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
49		fzonn0p1 13115	. . . . . 6 ⊢ (𝐸 ∈ ℕ₀ → 𝐸 ∈ (0..^(𝐸 + 1)))
50	48, 49	syl 17	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (0..^(𝐸 + 1)))
51		ccatws1len 13974	. . . . . . . 8 ⊢ ((𝑊 prefix 𝐸) ∈ Word 𝐷 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = ((♯‘(𝑊 prefix 𝐸)) + 1))
52	22, 23, 51	3syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = ((♯‘(𝑊 prefix 𝐸)) + 1))
53		pfxlen 14045	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
54	22, 47, 53	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
55	54	oveq1d 7171	. . . . . . 7 ⊢ (𝜑 → ((♯‘(𝑊 prefix 𝐸)) + 1) = (𝐸 + 1))
56	52, 55	eqtrd 2856	. . . . . 6 ⊢ (𝜑 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = (𝐸 + 1))
57	56	oveq2d 7172	. . . . 5 ⊢ (𝜑 → (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))) = (0..^(𝐸 + 1)))
58	50, 57	eleqtrrd 2916	. . . 4 ⊢ (𝜑 → 𝐸 ∈ (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))))
59		ccatval1 13930	. . . 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 1367	. . 3 ⊢ (𝜑 → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘𝐸) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘𝐸))
61	48	nn0zd 12086	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℤ)
62		elfzomin 13110	. . . . . 6 ⊢ (𝐸 ∈ ℤ → 𝐸 ∈ (𝐸..^(𝐸 + 1)))
63	61, 62	syl 17	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (𝐸..^(𝐸 + 1)))
64		s1len 13960	. . . . . . . 8 ⊢ (♯‘⟨“𝐼”⟩) = 1
65	64	a1i 11	. . . . . . 7 ⊢ (𝜑 → (♯‘⟨“𝐼”⟩) = 1)
66	54, 65	oveq12d 7174	. . . . . 6 ⊢ (𝜑 → ((♯‘(𝑊 prefix 𝐸)) + (♯‘⟨“𝐼”⟩)) = (𝐸 + 1))
67	54, 66	oveq12d 7174	. . . . 5 ⊢ (𝜑 → ((♯‘(𝑊 prefix 𝐸))..^((♯‘(𝑊 prefix 𝐸)) + (♯‘⟨“𝐼”⟩))) = (𝐸..^(𝐸 + 1)))
68	63, 67	eleqtrrd 2916	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ((♯‘(𝑊 prefix 𝐸))..^((♯‘(𝑊 prefix 𝐸)) + (♯‘⟨“𝐼”⟩))))
69		ccatval2 13932	. . . 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 1367	. . 3 ⊢ (𝜑 → (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘𝐸) = (⟨“𝐼”⟩‘(𝐸 − (♯‘(𝑊 prefix 𝐸)))))
71	12, 60, 70	3eqtrd 2860	. 2 ⊢ (𝜑 → (𝑈‘𝐸) = (⟨“𝐼”⟩‘(𝐸 − (♯‘(𝑊 prefix 𝐸)))))
72	54	oveq2d 7172	. . . 4 ⊢ (𝜑 → (𝐸 − (♯‘(𝑊 prefix 𝐸))) = (𝐸 − 𝐸))
73	48	nn0cnd 11958	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℂ)
74	73	subidd 10985	. . . 4 ⊢ (𝜑 → (𝐸 − 𝐸) = 0)
75	72, 74	eqtrd 2856	. . 3 ⊢ (𝜑 → (𝐸 − (♯‘(𝑊 prefix 𝐸))) = 0)
76	75	fveq2d 6674	. 2 ⊢ (𝜑 → (⟨“𝐼”⟩‘(𝐸 − (♯‘(𝑊 prefix 𝐸)))) = (⟨“𝐼”⟩‘0))
77		s1fv 13964	. . 3 ⊢ (𝐼 ∈ (𝐷 ∖ ran 𝑊) → (⟨“𝐼”⟩‘0) = 𝐼)
78	6, 77	syl 17	. 2 ⊢ (𝜑 → (⟨“𝐼”⟩‘0) = 𝐼)
79	71, 76, 78	3eqtrd 2860	1 ⊢ (𝜑 → (𝑈‘𝐸) = 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {crab 3142 Vcvv 3494 ∖ cdif 3933 ⟨cop 4573 ⟨cotp 4575 ^◡ccnv 5554 dom cdm 5555 ran crn 5556 ⟶wf 6351 –1-1→wf1 6352 –1-1-onto→wf1o 6354 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 + caddc 10540 − cmin 10870 ℕ₀cn0 11898 ℤcz 11982 ...cfz 12893 ..^cfzo 13034 ♯chash 13691 Word cword 13862 ++ cconcat 13922 ⟨“cs1 13949 substr csubstr 14002 prefix cpfx 14032 splice csplice 14111 Basecbs 16483 SymGrpcsymg 18495 toCycctocyc 30748

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-hash 13692 df-word 13863 df-concat 13923 df-s1 13950 df-substr 14003 df-pfx 14033 df-splice 14112 df-csh 14151 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-tset 16584 df-efmnd 18034 df-symg 18496 df-tocyc 30749

This theorem is referenced by: cycpmco2lem4 30771 cycpmco2lem5 30772 cycpmco2lem6 30773 cycpmco2 30775

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