fourierdlem52 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem52 43589

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 13095	. . . . . 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 43574	. . . 4 ⊢ (𝜑 → 𝑆 Isom < , < ((0...𝑁), 𝑇))
10		isof1o 7174	. . . 4 ⊢ (𝑆 Isom < , < ((0...𝑁), 𝑇) → 𝑆:(0...𝑁)–1-1-onto→𝑇)
11		f1of 6700	. . . 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 6602	. 2 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
14		f1ofo 6707	. . . . . 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 6964	. . . . 5 ⊢ ((𝑆:(0...𝑁)–onto→𝑇 ∧ 𝐴 ∈ 𝑇) → ∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆‘𝑗))
18	15, 16, 17	syl2anc 583	. . . 4 ⊢ (𝜑 → ∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆‘𝑗))
19		elfzle1 13188	. . . . . . . . 9 ⊢ (𝑗 ∈ (0...𝑁) → 0 ≤ 𝑗)
20	19	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 0 ≤ 𝑗)
21	9	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑆 Isom < , < ((0...𝑁), 𝑇))
22		ressxr 10950	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℝ^*
23	6, 22	sstrdi 3929	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ⊆ ℝ^*)
24	23	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑇 ⊆ ℝ^*)
25		fzssz 13187	. . . . . . . . . . 11 ⊢ (0...𝑁) ⊆ ℤ
26		zssre 12256	. . . . . . . . . . . 12 ⊢ ℤ ⊆ ℝ
27	26, 22	sstri 3926	. . . . . . . . . . 11 ⊢ ℤ ⊆ ℝ^*
28	25, 27	sstri 3926	. . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℝ^*
29	24, 28	jctil 519	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((0...𝑁) ⊆ ℝ^* ∧ 𝑇 ⊆ ℝ^*))
30		hashcl 13999	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ Fin → (♯‘𝑇) ∈ ℕ₀)
31	1, 30	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑇) ∈ ℕ₀)
32	16	ne0d 4266	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑇 ≠ ∅)
33		hashge1 14032	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ Fin ∧ 𝑇 ≠ ∅) → 1 ≤ (♯‘𝑇))
34	1, 32, 33	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ (♯‘𝑇))
35		elnnnn0c 12208	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑇) ∈ ℕ ↔ ((♯‘𝑇) ∈ ℕ₀ ∧ 1 ≤ (♯‘𝑇)))
36	31, 34, 35	sylanbrc 582	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑇) ∈ ℕ)
37		nnm1nn0 12204	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑇) ∈ ℕ → ((♯‘𝑇) − 1) ∈ ℕ₀)
38	36, 37	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((♯‘𝑇) − 1) ∈ ℕ₀)
39	8, 38	eqeltrid 2843	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
40		nn0uz 12549	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
41	39, 40	eleqtrdi 2849	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
42		eluzfz1 13192	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑁))
43	41, 42	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ (0...𝑁))
44	43	anim1i 614	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (0 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑁)))
45		leisorel 14102	. . . . . . . . 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 1369	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (0 ≤ 𝑗 ↔ (𝑆‘0) ≤ (𝑆‘𝑗)))
47	20, 46	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑆‘0) ≤ (𝑆‘𝑗))
48	47	3adant3 1130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆‘𝑗)) → (𝑆‘0) ≤ (𝑆‘𝑗))
49		eqcom 2745	. . . . . . . 8 ⊢ (𝐴 = (𝑆‘𝑗) ↔ (𝑆‘𝑗) = 𝐴)
50	49	biimpi 215	. . . . . . 7 ⊢ (𝐴 = (𝑆‘𝑗) → (𝑆‘𝑗) = 𝐴)
51	50	3ad2ant3 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆‘𝑗)) → (𝑆‘𝑗) = 𝐴)
52	48, 51	breqtrd 5096	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆‘𝑗)) → (𝑆‘0) ≤ 𝐴)
53	52	rexlimdv3a 3214	. . . 4 ⊢ (𝜑 → (∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆‘𝑗) → (𝑆‘0) ≤ 𝐴))
54	18, 53	mpd 15	. . 3 ⊢ (𝜑 → (𝑆‘0) ≤ 𝐴)
55	3	rexrd 10956	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
56	4	rexrd 10956	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
57	13, 43	ffvelrnd 6944	. . . 4 ⊢ (𝜑 → (𝑆‘0) ∈ (𝐴[,]𝐵))
58		iccgelb 13064	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑆‘0) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑆‘0))
59	55, 56, 57, 58	syl3anc 1369	. . 3 ⊢ (𝜑 → 𝐴 ≤ (𝑆‘0))
60	5, 57	sseldd 3918	. . . 4 ⊢ (𝜑 → (𝑆‘0) ∈ ℝ)
61	60, 3	letri3d 11047	. . 3 ⊢ (𝜑 → ((𝑆‘0) = 𝐴 ↔ ((𝑆‘0) ≤ 𝐴 ∧ 𝐴 ≤ (𝑆‘0))))
62	54, 59, 61	mpbir2and 709	. 2 ⊢ (𝜑 → (𝑆‘0) = 𝐴)
63		eluzfz2 13193	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 𝑁 ∈ (0...𝑁))
64	41, 63	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
65	13, 64	ffvelrnd 6944	. . . 4 ⊢ (𝜑 → (𝑆‘𝑁) ∈ (𝐴[,]𝐵))
66		iccleub 13063	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑆‘𝑁) ∈ (𝐴[,]𝐵)) → (𝑆‘𝑁) ≤ 𝐵)
67	55, 56, 65, 66	syl3anc 1369	. . 3 ⊢ (𝜑 → (𝑆‘𝑁) ≤ 𝐵)
68		fourierdlem52.bt	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑇)
69		foelrn 6964	. . . . 5 ⊢ ((𝑆:(0...𝑁)–onto→𝑇 ∧ 𝐵 ∈ 𝑇) → ∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆‘𝑗))
70	15, 68, 69	syl2anc 583	. . . 4 ⊢ (𝜑 → ∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆‘𝑗))
71		simp3 1136	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝐵 = (𝑆‘𝑗))
72		elfzle2 13189	. . . . . . . 8 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ≤ 𝑁)
73	72	3ad2ant2 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝑗 ≤ 𝑁)
74	9	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝑆 Isom < , < ((0...𝑁), 𝑇))
75	29	3adant3 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → ((0...𝑁) ⊆ ℝ^* ∧ 𝑇 ⊆ ℝ^*))
76		simp2 1135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝑗 ∈ (0...𝑁))
77	64	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝑁 ∈ (0...𝑁))
78		leisorel 14102	. . . . . . . 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 1372	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → (𝑗 ≤ 𝑁 ↔ (𝑆‘𝑗) ≤ (𝑆‘𝑁)))
80	73, 79	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → (𝑆‘𝑗) ≤ (𝑆‘𝑁))
81	71, 80	eqbrtrd 5092	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝐵 ≤ (𝑆‘𝑁))
82	81	rexlimdv3a 3214	. . . 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 11047	. . 3 ⊢ (𝜑 → ((𝑆‘𝑁) = 𝐵 ↔ ((𝑆‘𝑁) ≤ 𝐵 ∧ 𝐵 ≤ (𝑆‘𝑁))))
86	67, 83, 85	mpbir2and 709	. 2 ⊢ (𝜑 → (𝑆‘𝑁) = 𝐵)
87	13, 62, 86	jca31 514	1 ⊢ (𝜑 → ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (𝑆‘0) = 𝐴) ∧ (𝑆‘𝑁) = 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∃wrex 3064 ⊆ wss 3883 ∅c0 4253 class class class wbr 5070 ℩cio 6374 ⟶wf 6414 –onto→wfo 6416 –1-1-onto→wf1o 6417 ‘cfv 6418 Isom wiso 6419 (class class class)co 7255 Fincfn 8691 ℝcr 10801 0cc0 10802 1c1 10803 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 − cmin 11135 ℕcn 11903 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 [,]cicc 13011 ...cfz 13168 ♯chash 13972

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512 df-icc 13015 df-fz 13169 df-hash 13973

This theorem is referenced by: fourierdlem103 43640 fourierdlem104 43641

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