fourierdlem52 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem52 46986

Description: d16:d17,d18:jca |- ( ph -> ( ( S 0) ≤ 𝐴 ∧ 𝐴 ≤ (𝑆 0 ) ) ) . (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
fourierdlem52.tf	⊢ (𝜑 → 𝑇 ∈ Fin)
fourierdlem52.n	⊢ 𝑁 = ((♯‘𝑇) − 1)
fourierdlem52.s	⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))
fourierdlem52.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
fourierdlem52.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
fourierdlem52.t	⊢ (𝜑 → 𝑇 ⊆ (𝐴[,]𝐵))
fourierdlem52.at	⊢ (𝜑 → 𝐴 ∈ 𝑇)
fourierdlem52.bt	⊢ (𝜑 → 𝐵 ∈ 𝑇)

**Assertion**
Ref	Expression
fourierdlem52	⊢ (𝜑 → ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (𝑆‘0) = 𝐴) ∧ (𝑆‘𝑁) = 𝐵))

Distinct variable groups: 𝑓,𝑁 𝑆,𝑓 𝑇,𝑓 𝜑,𝑓 Allowed substitution hints: 𝐴(𝑓) 𝐵(𝑓)

Proof of Theorem fourierdlem52 Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem52.tf	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ Fin)
2		fourierdlem52.t	. . . . . 6 ⊢ (𝜑 → 𝑇 ⊆ (𝐴[,]𝐵))
3		fourierdlem52.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
4		fourierdlem52.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5	3, 4	iccssred 13487	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
6	2, 5	sstrd 3941	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ ℝ)
7		fourierdlem52.s	. . . . 5 ⊢ 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))
8		fourierdlem52.n	. . . . 5 ⊢ 𝑁 = ((♯‘𝑇) − 1)
9	1, 6, 7, 8	fourierdlem36 46971	. . . 4 ⊢ (𝜑 → 𝑆 Isom < , < ((0...𝑁), 𝑇))
10		isof1o 7324	. . . 4 ⊢ (𝑆 Isom < , < ((0...𝑁), 𝑇) → 𝑆:(0...𝑁)–1-1-onto→𝑇)
11		f1of 6817	. . . 4 ⊢ (𝑆:(0...𝑁)–1-1-onto→𝑇 → 𝑆:(0...𝑁)⟶𝑇)
12	9, 10, 11	3syl 19	. . 3 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶𝑇)
13	12, 2	fssd 6720	. 2 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
14		f1ofo 6825	. . . . . 6 ⊢ (𝑆:(0...𝑁)–1-1-onto→𝑇 → 𝑆:(0...𝑁)–onto→𝑇)
15	9, 10, 14	3syl 19	. . . . 5 ⊢ (𝜑 → 𝑆:(0...𝑁)–onto→𝑇)
16		fourierdlem52.at	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑇)
17		foelrn 7100	. . . . 5 ⊢ ((𝑆:(0...𝑁)–onto→𝑇 ∧ 𝐴 ∈ 𝑇) → ∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆‘𝑗))
18	15, 16, 17	syl2anc 596	. . . 4 ⊢ (𝜑 → ∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆‘𝑗))
19		elfzle1 13581	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑁) → 0 ≤ 𝑗)
20	19	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 0 ≤ 𝑗)
21	9	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑆 Isom < , < ((0...𝑁), 𝑇))
22		ressxr 11277	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℝ^*
23	6, 22	sstrdi 3943	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ⊆ ℝ^*)
24	23	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑇 ⊆ ℝ^*)
25		fzssz 13580	. . . . . . . . . . 11 ⊢ (0...𝑁) ⊆ ℤ
26		zssre 12622	. . . . . . . . . . . 12 ⊢ ℤ ⊆ ℝ
27	26, 22	sstri 3940	. . . . . . . . . . 11 ⊢ ℤ ⊆ ℝ^*
28	25, 27	sstri 3940	. . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℝ^*
29	24, 28	jctil 529	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((0...𝑁) ⊆ ℝ^* ∧ 𝑇 ⊆ ℝ^*))
30		hashcl 14420	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ Fin → (♯‘𝑇) ∈ ℕ₀)
31	1, 30	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑇) ∈ ℕ₀)
32	16	ne0d 4288	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑇 ≠ ∅)
33		hashge1 14453	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ Fin ∧ 𝑇 ≠ ∅) → 1 ≤ (♯‘𝑇))
34	1, 32, 33	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ (♯‘𝑇))
35		elnnnn0c 12573	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑇) ∈ ℕ ↔ ((♯‘𝑇) ∈ ℕ₀ ∧ 1 ≤ (♯‘𝑇)))
36	31, 34, 35	sylanbrc 595	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑇) ∈ ℕ)
37		nnm1nn0 12569	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑇) ∈ ℕ → ((♯‘𝑇) − 1) ∈ ℕ₀)
38	36, 37	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((♯‘𝑇) − 1) ∈ ℕ₀)
39	8, 38	eqeltrid 2864	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
40		nn0uz 12925	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
41	39, 40	eleqtrdi 2870	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
42		eluzfz1 13585	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑁))
43	41, 42	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ (0...𝑁))
44	43	anim1i 627	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (0 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑁)))
45		leisorel 14525	. . . . . . . . 9 ⊢ ((𝑆 Isom < , < ((0...𝑁), 𝑇) ∧ ((0...𝑁) ⊆ ℝ^* ∧ 𝑇 ⊆ ℝ^*) ∧ (0 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑁))) → (0 ≤ 𝑗 ↔ (𝑆‘0) ≤ (𝑆‘𝑗)))
46	21, 29, 44, 45	syl3anc 1398	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (0 ≤ 𝑗 ↔ (𝑆‘0) ≤ (𝑆‘𝑗)))
47	20, 46	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑆‘0) ≤ (𝑆‘𝑗))
48	47	3adant3 1150	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆‘𝑗)) → (𝑆‘0) ≤ (𝑆‘𝑗))
49		eqcom 2767	. . . . . . . 8 ⊢ (𝐴 = (𝑆‘𝑗) ↔ (𝑆‘𝑗) = 𝐴)
50	49	biimpi 219	. . . . . . 7 ⊢ (𝐴 = (𝑆‘𝑗) → (𝑆‘𝑗) = 𝐴)
51	50	3ad2ant3 1153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆‘𝑗)) → (𝑆‘𝑗) = 𝐴)
52	48, 51	breqtrd 5131	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆‘𝑗)) → (𝑆‘0) ≤ 𝐴)
53	52	rexlimdv3a 3167	. . . 4 ⊢ (𝜑 → (∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆‘𝑗) → (𝑆‘0) ≤ 𝐴))
54	18, 53	mpd 16	. . 3 ⊢ (𝜑 → (𝑆‘0) ≤ 𝐴)
55	3	rexrd 11283	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
56	4	rexrd 11283	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
57	13, 43	ffvelcdmd 7078	. . . 4 ⊢ (𝜑 → (𝑆‘0) ∈ (𝐴[,]𝐵))
58		iccgelb 13455	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑆‘0) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑆‘0))
59	55, 56, 57, 58	syl3anc 1398	. . 3 ⊢ (𝜑 → 𝐴 ≤ (𝑆‘0))
60	5, 57	sseldd 3932	. . . 4 ⊢ (𝜑 → (𝑆‘0) ∈ ℝ)
61	60, 3	letri3d 11376	. . 3 ⊢ (𝜑 → ((𝑆‘0) = 𝐴 ↔ ((𝑆‘0) ≤ 𝐴 ∧ 𝐴 ≤ (𝑆‘0))))
62	54, 59, 61	mpbir2and 726	. 2 ⊢ (𝜑 → (𝑆‘0) = 𝐴)
63		eluzfz2 13586	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 𝑁 ∈ (0...𝑁))
64	41, 63	syl 18	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
65	13, 64	ffvelcdmd 7078	. . . 4 ⊢ (𝜑 → (𝑆‘𝑁) ∈ (𝐴[,]𝐵))
66		iccleub 13454	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑆‘𝑁) ∈ (𝐴[,]𝐵)) → (𝑆‘𝑁) ≤ 𝐵)
67	55, 56, 65, 66	syl3anc 1398	. . 3 ⊢ (𝜑 → (𝑆‘𝑁) ≤ 𝐵)
68		fourierdlem52.bt	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑇)
69		foelrn 7100	. . . . 5 ⊢ ((𝑆:(0...𝑁)–onto→𝑇 ∧ 𝐵 ∈ 𝑇) → ∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆‘𝑗))
70	15, 68, 69	syl2anc 596	. . . 4 ⊢ (𝜑 → ∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆‘𝑗))
71		simp3 1156	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝐵 = (𝑆‘𝑗))
72		elfzle2 13582	. . . . . . . 8 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ≤ 𝑁)
73	72	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝑗 ≤ 𝑁)
74	9	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝑆 Isom < , < ((0...𝑁), 𝑇))
75	29	3adant3 1150	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → ((0...𝑁) ⊆ ℝ^* ∧ 𝑇 ⊆ ℝ^*))
76		simp2 1155	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝑗 ∈ (0...𝑁))
77	64	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝑁 ∈ (0...𝑁))
78		leisorel 14525	. . . . . . . 8 ⊢ ((𝑆 Isom < , < ((0...𝑁), 𝑇) ∧ ((0...𝑁) ⊆ ℝ^* ∧ 𝑇 ⊆ ℝ^*) ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...𝑁))) → (𝑗 ≤ 𝑁 ↔ (𝑆‘𝑗) ≤ (𝑆‘𝑁)))
79	74, 75, 76, 77, 78	syl112anc 1401	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → (𝑗 ≤ 𝑁 ↔ (𝑆‘𝑗) ≤ (𝑆‘𝑁)))
80	73, 79	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → (𝑆‘𝑗) ≤ (𝑆‘𝑁))
81	71, 80	eqbrtrd 5127	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝐵 ≤ (𝑆‘𝑁))
82	81	rexlimdv3a 3167	. . . 4 ⊢ (𝜑 → (∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆‘𝑗) → 𝐵 ≤ (𝑆‘𝑁)))
83	70, 82	mpd 16	. . 3 ⊢ (𝜑 → 𝐵 ≤ (𝑆‘𝑁))
84	5, 65	sseldd 3932	. . . 4 ⊢ (𝜑 → (𝑆‘𝑁) ∈ ℝ)
85	84, 4	letri3d 11376	. . 3 ⊢ (𝜑 → ((𝑆‘𝑁) = 𝐵 ↔ ((𝑆‘𝑁) ≤ 𝐵 ∧ 𝐵 ≤ (𝑆‘𝑁))))
86	67, 83, 85	mpbir2and 726	. 2 ⊢ (𝜑 → (𝑆‘𝑁) = 𝐵)
87	13, 62, 86	jca31 524	1 ⊢ (𝜑 → ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (𝑆‘0) = 𝐴) ∧ (𝑆‘𝑁) = 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∃wrex 3086 ⊆ wss 3899 ∅c0 4279 class class class wbr 5103 ℩cio 6487 ⟶wf 6529 –onto→wfo 6531 –1-1-onto→wf1o 6532 ‘cfv 6533 Isom wiso 6534 (class class class)co 7413 Fincfn 8952 ℝcr 11123 0cc0 11124 1c1 11125 ℝ^*cxr 11266 < clt 11267 ≤ cle 11268 − cmin 11465 ℕcn 12257 ℕ₀cn0 12528 ℤcz 12615 ℤ_≥cuz 12887 [,]cicc 13401 ...cfz 13561 ♯chash 14394

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-oi 9482 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-n0 12529 df-z 12616 df-uz 12888 df-icc 13405 df-fz 13562 df-hash 14395

This theorem is used by: fourierdlem103 47037 fourierdlem104 47038

W3C validator