fourierdlem52 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem52 46615

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 13382	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
6	2, 5	sstrd 3927	. . . . 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 46600	. . . 4 ⊢ (𝜑 → 𝑆 Isom < , < ((0...𝑁), 𝑇))
10		isof1o 7271	. . . 4 ⊢ (𝑆 Isom < , < ((0...𝑁), 𝑇) → 𝑆:(0...𝑁)–1-1-onto→𝑇)
11		f1of 6771	. . . 4 ⊢ (𝑆:(0...𝑁)–1-1-onto→𝑇 → 𝑆:(0...𝑁)⟶𝑇)
12	9, 10, 11	3syl 18	. . 3 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶𝑇)
13	12, 2	fssd 6676	. 2 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
14		f1ofo 6778	. . . . . 6 ⊢ (𝑆:(0...𝑁)–1-1-onto→𝑇 → 𝑆:(0...𝑁)–onto→𝑇)
15	9, 10, 14	3syl 18	. . . . 5 ⊢ (𝜑 → 𝑆:(0...𝑁)–onto→𝑇)
16		fourierdlem52.at	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑇)
17		foelrn 7052	. . . . 5 ⊢ ((𝑆:(0...𝑁)–onto→𝑇 ∧ 𝐴 ∈ 𝑇) → ∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆‘𝑗))
18	15, 16, 17	syl2anc 591	. . . 4 ⊢ (𝜑 → ∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆‘𝑗))
19		elfzle1 13476	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑁) → 0 ≤ 𝑗)
20	19	adantl 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 0 ≤ 𝑗)
21	9	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑆 Isom < , < ((0...𝑁), 𝑇))
22		ressxr 11184	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℝ^*
23	6, 22	sstrdi 3929	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ⊆ ℝ^*)
24	23	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑇 ⊆ ℝ^*)
25		fzssz 13475	. . . . . . . . . . 11 ⊢ (0...𝑁) ⊆ ℤ
26		zssre 12526	. . . . . . . . . . . 12 ⊢ ℤ ⊆ ℝ
27	26, 22	sstri 3926	. . . . . . . . . . 11 ⊢ ℤ ⊆ ℝ^*
28	25, 27	sstri 3926	. . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℝ^*
29	24, 28	jctil 525	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((0...𝑁) ⊆ ℝ^* ∧ 𝑇 ⊆ ℝ^*))
30		hashcl 14313	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ Fin → (♯‘𝑇) ∈ ℕ₀)
31	1, 30	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑇) ∈ ℕ₀)
32	16	ne0d 4273	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑇 ≠ ∅)
33		hashge1 14346	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ Fin ∧ 𝑇 ≠ ∅) → 1 ≤ (♯‘𝑇))
34	1, 32, 33	syl2anc 591	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ (♯‘𝑇))
35		elnnnn0c 12477	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑇) ∈ ℕ ↔ ((♯‘𝑇) ∈ ℕ₀ ∧ 1 ≤ (♯‘𝑇)))
36	31, 34, 35	sylanbrc 590	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑇) ∈ ℕ)
37		nnm1nn0 12473	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑇) ∈ ℕ → ((♯‘𝑇) − 1) ∈ ℕ₀)
38	36, 37	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((♯‘𝑇) − 1) ∈ ℕ₀)
39	8, 38	eqeltrid 2845	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
40		nn0uz 12821	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
41	39, 40	eleqtrdi 2851	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
42		eluzfz1 13480	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑁))
43	41, 42	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ (0...𝑁))
44	43	anim1i 622	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (0 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑁)))
45		leisorel 14417	. . . . . . . . 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 1380	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (0 ≤ 𝑗 ↔ (𝑆‘0) ≤ (𝑆‘𝑗)))
47	20, 46	mpbid 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑆‘0) ≤ (𝑆‘𝑗))
48	47	3adant3 1139	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆‘𝑗)) → (𝑆‘0) ≤ (𝑆‘𝑗))
49		eqcom 2748	. . . . . . . 8 ⊢ (𝐴 = (𝑆‘𝑗) ↔ (𝑆‘𝑗) = 𝐴)
50	49	biimpi 218	. . . . . . 7 ⊢ (𝐴 = (𝑆‘𝑗) → (𝑆‘𝑗) = 𝐴)
51	50	3ad2ant3 1142	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆‘𝑗)) → (𝑆‘𝑗) = 𝐴)
52	48, 51	breqtrd 5101	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆‘𝑗)) → (𝑆‘0) ≤ 𝐴)
53	52	rexlimdv3a 3146	. . . 4 ⊢ (𝜑 → (∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆‘𝑗) → (𝑆‘0) ≤ 𝐴))
54	18, 53	mpd 15	. . 3 ⊢ (𝜑 → (𝑆‘0) ≤ 𝐴)
55	3	rexrd 11190	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
56	4	rexrd 11190	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
57	13, 43	ffvelcdmd 7030	. . . 4 ⊢ (𝜑 → (𝑆‘0) ∈ (𝐴[,]𝐵))
58		iccgelb 13350	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑆‘0) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑆‘0))
59	55, 56, 57, 58	syl3anc 1380	. . 3 ⊢ (𝜑 → 𝐴 ≤ (𝑆‘0))
60	5, 57	sseldd 3918	. . . 4 ⊢ (𝜑 → (𝑆‘0) ∈ ℝ)
61	60, 3	letri3d 11283	. . 3 ⊢ (𝜑 → ((𝑆‘0) = 𝐴 ↔ ((𝑆‘0) ≤ 𝐴 ∧ 𝐴 ≤ (𝑆‘0))))
62	54, 59, 61	mpbir2and 720	. 2 ⊢ (𝜑 → (𝑆‘0) = 𝐴)
63		eluzfz2 13481	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 𝑁 ∈ (0...𝑁))
64	41, 63	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
65	13, 64	ffvelcdmd 7030	. . . 4 ⊢ (𝜑 → (𝑆‘𝑁) ∈ (𝐴[,]𝐵))
66		iccleub 13349	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑆‘𝑁) ∈ (𝐴[,]𝐵)) → (𝑆‘𝑁) ≤ 𝐵)
67	55, 56, 65, 66	syl3anc 1380	. . 3 ⊢ (𝜑 → (𝑆‘𝑁) ≤ 𝐵)
68		fourierdlem52.bt	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑇)
69		foelrn 7052	. . . . 5 ⊢ ((𝑆:(0...𝑁)–onto→𝑇 ∧ 𝐵 ∈ 𝑇) → ∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆‘𝑗))
70	15, 68, 69	syl2anc 591	. . . 4 ⊢ (𝜑 → ∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆‘𝑗))
71		simp3 1145	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝐵 = (𝑆‘𝑗))
72		elfzle2 13477	. . . . . . . 8 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ≤ 𝑁)
73	72	3ad2ant2 1141	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝑗 ≤ 𝑁)
74	9	3ad2ant1 1140	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝑆 Isom < , < ((0...𝑁), 𝑇))
75	29	3adant3 1139	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → ((0...𝑁) ⊆ ℝ^* ∧ 𝑇 ⊆ ℝ^*))
76		simp2 1144	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝑗 ∈ (0...𝑁))
77	64	3ad2ant1 1140	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝑁 ∈ (0...𝑁))
78		leisorel 14417	. . . . . . . 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 1383	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → (𝑗 ≤ 𝑁 ↔ (𝑆‘𝑗) ≤ (𝑆‘𝑁)))
80	73, 79	mpbid 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → (𝑆‘𝑗) ≤ (𝑆‘𝑁))
81	71, 80	eqbrtrd 5097	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝐵 ≤ (𝑆‘𝑁))
82	81	rexlimdv3a 3146	. . . 4 ⊢ (𝜑 → (∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆‘𝑗) → 𝐵 ≤ (𝑆‘𝑁)))
83	70, 82	mpd 15	. . 3 ⊢ (𝜑 → 𝐵 ≤ (𝑆‘𝑁))
84	5, 65	sseldd 3918	. . . 4 ⊢ (𝜑 → (𝑆‘𝑁) ∈ ℝ)
85	84, 4	letri3d 11283	. . 3 ⊢ (𝜑 → ((𝑆‘𝑁) = 𝐵 ↔ ((𝑆‘𝑁) ≤ 𝐵 ∧ 𝐵 ≤ (𝑆‘𝑁))))
86	67, 83, 85	mpbir2and 720	. 2 ⊢ (𝜑 → (𝑆‘𝑁) = 𝐵)
87	13, 62, 86	jca31 520	1 ⊢ (𝜑 → ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (𝑆‘0) = 𝐴) ∧ (𝑆‘𝑁) = 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 397 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∃wrex 3065 ⊆ wss 3885 ∅c0 4264 class class class wbr 5075 ℩cio 6443 ⟶wf 6485 –onto→wfo 6487 –1-1-onto→wf1o 6488 ‘cfv 6489 Isom wiso 6490 (class class class)co 7360 Fincfn 8887 ℝcr 11032 0cc0 11033 1c1 11034 ℝ^*cxr 11173 < clt 11174 ≤ cle 11175 − cmin 11372 ℕcn 12169 ℕ₀cn0 12432 ℤcz 12519 ℤ_≥cuz 12783 [,]cicc 13296 ...cfz 13456 ♯chash 14287

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-inf2 9557 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-oi 9419 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-n0 12433 df-z 12520 df-uz 12784 df-icc 13300 df-fz 13457 df-hash 14288

This theorem is referenced by: fourierdlem103 46666 fourierdlem104 46667

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