cycpmco2lem2 - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > cycpmco2lem2		Structured version Visualization version GIF version

Theorem cycpmco2lem2 33103

Description: Lemma for cycpmco2 33109. (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 7387	. . . . . . 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 3910	. . . . . . 7 ⊢ (𝜑 → 𝐼 ∈ 𝐷)
8	7	s1cld 14513	. . . . . 6 ⊢ (𝜑 → ⟨“𝐼”⟩ ∈ Word 𝐷)
9		splval 14660	. . . . . 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 1374	. . . . 5 ⊢ (𝜑 → (𝑊 splice ⟨𝐸, 𝐸, ⟨“𝐼”⟩⟩) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
11	1, 10	eqtrid 2780	. . . 4 ⊢ (𝜑 → 𝑈 = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩)))
12	11	fveq1d 6830	. . 3 ⊢ (𝜑 → (𝑈‘𝐸) = ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘𝐸))
13		ssrab2 4029	. . . . . . 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 2733	. . . . . . . . . . 11 ⊢ (Base‘𝑆) = (Base‘𝑆)
18	15, 16, 17	tocycf 33093	. . . . . . . . . 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 6666	. . . . . . . 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 3928	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
23		pfxcl 14587	. . . . . 6 ⊢ (𝑊 ∈ Word 𝐷 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
24	22, 23	syl 17	. . . . 5 ⊢ (𝜑 → (𝑊 prefix 𝐸) ∈ Word 𝐷)
25		ccatcl 14483	. . . . 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 14555	. . . . 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 13524	. . . . . . 7 ⊢ (0...(♯‘𝑊)) ⊆ ℕ₀
30		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
31		dmeq 5847	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → dom 𝑤 = dom 𝑊)
32		eqidd 2734	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑊 → 𝐷 = 𝐷)
33	30, 31, 32	f1eq123d 6760	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑊 → (𝑤:dom 𝑤–1-1→𝐷 ↔ 𝑊:dom 𝑊–1-1→𝐷))
34	33	elrab 3643	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ {𝑤 ∈ Word 𝐷 ∣ 𝑤:dom 𝑤–1-1→𝐷} ↔ (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
35	21, 34	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑊 ∈ Word 𝐷 ∧ 𝑊:dom 𝑊–1-1→𝐷))
36	35	simprd 495	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
37		f1cnv 6792	. . . . . . . . . . . 12 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → ^◡𝑊:ran 𝑊–1-1-onto→dom 𝑊)
38		f1of 6768	. . . . . . . . . . . 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 7024	. . . . . . . . . 10 ⊢ (𝜑 → (^◡𝑊‘𝐽) ∈ dom 𝑊)
42		wrddm 14430	. . . . . . . . . . 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 13666	. . . . . . . . 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 3928	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℕ₀)
49		fzonn0p1 13644	. . . . . 6 ⊢ (𝐸 ∈ ℕ₀ → 𝐸 ∈ (0..^(𝐸 + 1)))
50	48, 49	syl 17	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (0..^(𝐸 + 1)))
51		ccatws1len 14530	. . . . . . . 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 14593	. . . . . . . . 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 7367	. . . . . . 7 ⊢ (𝜑 → ((♯‘(𝑊 prefix 𝐸)) + 1) = (𝐸 + 1))
56	52, 55	eqtrd 2768	. . . . . 6 ⊢ (𝜑 → (♯‘((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)) = (𝐸 + 1))
57	56	oveq2d 7368	. . . . 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 14486	. . . 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 1373	. . 3 ⊢ (𝜑 → ((((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩) ++ (𝑊 substr ⟨𝐸, (♯‘𝑊)⟩))‘𝐸) = (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘𝐸))
61	48	nn0zd 12500	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ ℤ)
62		elfzomin 13639	. . . . . 6 ⊢ (𝐸 ∈ ℤ → 𝐸 ∈ (𝐸..^(𝐸 + 1)))
63	61, 62	syl 17	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ (𝐸..^(𝐸 + 1)))
64		s1len 14516	. . . . . . . 8 ⊢ (♯‘⟨“𝐼”⟩) = 1
65	64	a1i 11	. . . . . . 7 ⊢ (𝜑 → (♯‘⟨“𝐼”⟩) = 1)
66	54, 65	oveq12d 7370	. . . . . 6 ⊢ (𝜑 → ((♯‘(𝑊 prefix 𝐸)) + (♯‘⟨“𝐼”⟩)) = (𝐸 + 1))
67	54, 66	oveq12d 7370	. . . . 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 14487	. . . 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 1373	. . 3 ⊢ (𝜑 → (((𝑊 prefix 𝐸) ++ ⟨“𝐼”⟩)‘𝐸) = (⟨“𝐼”⟩‘(𝐸 − (♯‘(𝑊 prefix 𝐸)))))
71	12, 60, 70	3eqtrd 2772	. 2 ⊢ (𝜑 → (𝑈‘𝐸) = (⟨“𝐼”⟩‘(𝐸 − (♯‘(𝑊 prefix 𝐸)))))
72	54	oveq2d 7368	. . . 4 ⊢ (𝜑 → (𝐸 − (♯‘(𝑊 prefix 𝐸))) = (𝐸 − 𝐸))
73	48	nn0cnd 12451	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℂ)
74	73	subidd 11467	. . . 4 ⊢ (𝜑 → (𝐸 − 𝐸) = 0)
75	72, 74	eqtrd 2768	. . 3 ⊢ (𝜑 → (𝐸 − (♯‘(𝑊 prefix 𝐸))) = 0)
76	75	fveq2d 6832	. 2 ⊢ (𝜑 → (⟨“𝐼”⟩‘(𝐸 − (♯‘(𝑊 prefix 𝐸)))) = (⟨“𝐼”⟩‘0))
77		s1fv 14520	. . 3 ⊢ (𝐼 ∈ (𝐷 ∖ ran 𝑊) → (⟨“𝐼”⟩‘0) = 𝐼)
78	6, 77	syl 17	. 2 ⊢ (𝜑 → (⟨“𝐼”⟩‘0) = 𝐼)
79	71, 76, 78	3eqtrd 2772	1 ⊢ (𝜑 → (𝑈‘𝐸) = 𝐼)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 {crab 3396 Vcvv 3437 ∖ cdif 3895 ⟨cop 4581 ⟨cotp 4583 ^◡ccnv 5618 dom cdm 5619 ran crn 5620 ⟶wf 6482 –1-1→wf1 6483 –1-1-onto→wf1o 6485 ‘cfv 6486 (class class class)co 7352 0cc0 11013 1c1 11014 + caddc 11016 − cmin 11351 ℕ₀cn0 12388 ℤcz 12475 ...cfz 13409 ..^cfzo 13556 ♯chash 14239 Word cword 14422 ++ cconcat 14479 ⟨“cs1 14505 substr csubstr 14550 prefix cpfx 14580 splice csplice 14658 Basecbs 17122 SymGrpcsymg 19283 toCycctocyc 33082

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-ot 4584 df-uni 4859 df-int 4898 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-er 8628 df-map 8758 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-sup 9333 df-inf 9334 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-uz 12739 df-rp 12893 df-fz 13410 df-fzo 13557 df-fl 13698 df-mod 13776 df-hash 14240 df-word 14423 df-concat 14480 df-s1 14506 df-substr 14551 df-pfx 14581 df-splice 14659 df-csh 14698 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-tset 17182 df-efmnd 18779 df-symg 19284 df-tocyc 33083

This theorem is referenced by: cycpmco2lem4 33105 cycpmco2lem5 33106 cycpmco2lem6 33107 cycpmco2 33109

W3C validator