cycpmco2lem2 - Mathbox for Thierry Arnoux

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Theorem cycpmco2lem2 32327

Description: Lemma for cycpmco2 32333. (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 7446	. . . . . . 7 ⊢ (𝜑 → ((^◡𝑊‘𝐽) + 1) ∈ V)
5	3, 4	eqeltrid 2837	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ V)
6		cycpmco2.i	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ (𝐷 ∖ ran 𝑊))
7	6	eldifad 3960	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
8	7	s1cld 14555	. . . . . 6 ⊢ (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
9		splval 14703	. . . . . 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 1372	. . . . 5 ⊢ (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
11	1, 10	eqtrid 2784	. . . 4 ⊢ (𝜑 → 𝑈 = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
12	11	fveq1d 6893	. . 3 ⊢ (𝜑 → (𝑈‘𝐸) = ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘𝐸))
13		ssrab2 4077	. . . . . . 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 2732	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
18	15, 16, 17	tocycf 32317	. . . . . . . . . 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 6728	. . . . . . . 8 ⊢ (𝜑 → dom 𝑀 = {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
21	2, 20	eleqtrd 2835	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷})
22	13, 21	sselid 3980	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
23		pfxcl 14629	. . . . . 6 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
24	22, 23	syl 17	. . . . 5 ⊢ (𝜑 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
25		ccatcl 14526	. . . . 5 ⊢ (((𝑊 prefix 𝐸) ∈ Word 𝐷 ∧ ⟨“𝐼”⟩ ∈ Word 𝐷) → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
26	24, 8, 25	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ∈ Word 𝐷)
27		swrdcl 14597	. . . . 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 13598	. . . . . . 7 ⊢ (0...(♯‘𝑊)) ⊆ ℕ₀
30		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
31		dmeq 5903	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
32		eqidd 2733	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷)
33	30, 31, 32	f1eq123d 6825	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷))
34	33	elrab 3683	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
35	21, 34	sylib 217	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
36	35	simprd 496	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
37		f1cnv 6857	. . . . . . . . . . . 12 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → ^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊)
38		f1of 6833	. . . . . . . . . . . 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	ffvelcdmd 7087	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ dom 𝑊)
42		wrddm 14473	. . . . . . . . . . 11 ⊢ (𝑊 ∈ Word 𝐷 → dom 𝑊 = (0..^(♯‘𝑊)))
43	22, 42	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
44	41, 43	eleqtrd 2835	. . . . . . . . 9 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ (0..^(♯‘𝑊)))
45		fzofzp1 13731	. . . . . . . . 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 2837	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (0...(♯‘𝑊)))
48	29, 47	sselid 3980	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
49		fzonn0p1 13711	. . . . . 6 ⊢ (𝐸 ∈ ℕ₀ → 𝐸 ∈ (0..^(𝐸 + 1)))
50	48, 49	syl 17	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (0..^(𝐸 + 1)))
51		ccatws1len 14572	. . . . . . . 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 14635	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 𝐸 ∈ (0...(♯‘𝑊))) → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
54	22, 47, 53	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (♯‘(𝑊 prefix 𝐸)) = 𝐸)
55	54	oveq1d 7426	. . . . . . 7 ⊢ (𝜑 → ((♯‘(𝑊 prefix 𝐸)) + 1) = (𝐸 + 1))
56	52, 55	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = (𝐸 + 1))
57	56	oveq2d 7427	. . . . 5 ⊢ (𝜑 → (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))) = (0..^(𝐸 + 1)))
58	50, 57	eleqtrrd 2836	. . . 4 ⊢ (𝜑 → 𝐸 ∈ (0..^(♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩))))
59		ccatval1 14529	. . . 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 1371	. . 3 ⊢ (𝜑 → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘𝐸) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘𝐸))
61	48	nn0zd 12586	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℤ)
62		elfzomin 13706	. . . . . 6 ⊢ (𝐸 ∈ ℤ → 𝐸 ∈ (𝐸..^(𝐸 + 1)))
63	61, 62	syl 17	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (𝐸..^(𝐸 + 1)))
64		s1len 14558	. . . . . . . 8 ⊢ (♯‘⟨“𝐼”⟩) = 1
65	64	a1i 11	. . . . . . 7 ⊢ (𝜑 → (♯‘⟨“𝐼”⟩) = 1)
66	54, 65	oveq12d 7429	. . . . . 6 ⊢ (𝜑 → ((♯‘(𝑊 prefix 𝐸)) + (♯‘⟨“𝐼”⟩)) = (𝐸 + 1))
67	54, 66	oveq12d 7429	. . . . 5 ⊢ (𝜑 → ((♯‘(𝑊 prefix 𝐸))..^((♯‘(𝑊 prefix 𝐸)) + (♯‘⟨“𝐼”⟩))) = (𝐸..^(𝐸 + 1)))
68	63, 67	eleqtrrd 2836	. . . 4 ⊢ (𝜑 → 𝐸 ∈ ((♯‘(𝑊 prefix 𝐸))..^((♯‘(𝑊 prefix 𝐸)) + (♯‘⟨“𝐼”⟩))))
69		ccatval2 14530	. . . 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 1371	. . 3 ⊢ (𝜑 → (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘𝐸) = (⟨“𝐼”⟩‘(𝐸 − (♯‘(𝑊 prefix 𝐸)))))
71	12, 60, 70	3eqtrd 2776	. 2 ⊢ (𝜑 → (𝑈‘𝐸) = (⟨“𝐼”⟩‘(𝐸 − (♯‘(𝑊 prefix 𝐸)))))
72	54	oveq2d 7427	. . . 4 ⊢ (𝜑 → (𝐸 − (♯‘(𝑊 prefix 𝐸))) = (𝐸 − 𝐸))
73	48	nn0cnd 12536	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℂ)
74	73	subidd 11561	. . . 4 ⊢ (𝜑 → (𝐸 − 𝐸) = 0)
75	72, 74	eqtrd 2772	. . 3 ⊢ (𝜑 → (𝐸 − (♯‘(𝑊 prefix 𝐸))) = 0)
76	75	fveq2d 6895	. 2 ⊢ (𝜑 → (⟨“𝐼”⟩‘(𝐸 − (♯‘(𝑊 prefix 𝐸)))) = (⟨“𝐼”⟩‘0))
77		s1fv 14562	. . 3 ⊢ (𝐼 ∈ (𝐷 ∖ ran 𝑊) → (⟨“𝐼”⟩‘0) = 𝐼)
78	6, 77	syl 17	. 2 ⊢ (𝜑 → (⟨“𝐼”⟩‘0) = 𝐼)
79	71, 76, 78	3eqtrd 2776	1 ⊢ (𝜑 → (𝑈‘𝐸) = 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3432 Vcvv 3474 ∖ cdif 3945 ⟨cop 4634 ⟨cotp 4636 ^◡ccnv 5675 dom cdm 5676 ran crn 5677 ⟶wf 6539 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7411 0cc0 11112 1c1 11113 + caddc 11115 − cmin 11446 ℕ₀cn0 12474 ℤcz 12560 ...cfz 13486 ..^cfzo 13629 ♯chash 14292 Word cword 14466 ++ cconcat 14522 ⟨“cs1 14547 substr csubstr 14592 prefix cpfx 14622 splice csplice 14701 Basecbs 17146 SymGrpcsymg 19236 toCycctocyc 32306

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-ot 4637 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-uz 12825 df-rp 12977 df-fz 13487 df-fzo 13630 df-fl 13759 df-mod 13837 df-hash 14293 df-word 14467 df-concat 14523 df-s1 14548 df-substr 14593 df-pfx 14623 df-splice 14702 df-csh 14741 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-tset 17218 df-efmnd 18752 df-symg 19237 df-tocyc 32307

This theorem is referenced by: cycpmco2lem4 32329 cycpmco2lem5 32330 cycpmco2lem6 32331 cycpmco2 32333

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