fourierdlem52 - Mathbox for Glauco Siliprandi

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Theorem fourierdlem52 46402

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 13350	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
6	2, 5	sstrd 3944	. . . . 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 46387	. . . 4 ⊢ (𝜑 → 𝑆 Isom < , < ((0...𝑁), 𝑇))
10		isof1o 7269	. . . 4 ⊢ (𝑆 Isom < , < ((0...𝑁), 𝑇) → 𝑆:(0...𝑁)–1-1-onto→𝑇)
11		f1of 6774	. . . 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 6679	. 2 ⊢ (𝜑 → 𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
14		f1ofo 6781	. . . . . 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 584	. . . 4 ⊢ (𝜑 → ∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆‘𝑗))
19		elfzle1 13443	. . . . . . . . 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 11176	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℝ^*
23	6, 22	sstrdi 3946	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ⊆ ℝ^*)
24	23	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → 𝑇 ⊆ ℝ^*)
25		fzssz 13442	. . . . . . . . . . 11 ⊢ (0...𝑁) ⊆ ℤ
26		zssre 12495	. . . . . . . . . . . 12 ⊢ ℤ ⊆ ℝ
27	26, 22	sstri 3943	. . . . . . . . . . 11 ⊢ ℤ ⊆ ℝ^*
28	25, 27	sstri 3943	. . . . . . . . . 10 ⊢ (0...𝑁) ⊆ ℝ^*
29	24, 28	jctil 519	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → ((0...𝑁) ⊆ ℝ^* ∧ 𝑇 ⊆ ℝ^*))
30		hashcl 14279	. . . . . . . . . . . . . . . 16 ⊢ (𝑇 ∈ Fin → (♯‘𝑇) ∈ ℕ₀)
31	1, 30	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (♯‘𝑇) ∈ ℕ₀)
32	16	ne0d 4294	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑇 ≠ ∅)
33		hashge1 14312	. . . . . . . . . . . . . . . 16 ⊢ ((𝑇 ∈ Fin ∧ 𝑇 ≠ ∅) → 1 ≤ (♯‘𝑇))
34	1, 32, 33	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ≤ (♯‘𝑇))
35		elnnnn0c 12446	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑇) ∈ ℕ ↔ ((♯‘𝑇) ∈ ℕ₀ ∧ 1 ≤ (♯‘𝑇)))
36	31, 34, 35	sylanbrc 583	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝑇) ∈ ℕ)
37		nnm1nn0 12442	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑇) ∈ ℕ → ((♯‘𝑇) − 1) ∈ ℕ₀)
38	36, 37	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((♯‘𝑇) − 1) ∈ ℕ₀)
39	8, 38	eqeltrid 2840	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
40		nn0uz 12789	. . . . . . . . . . . 12 ⊢ ℕ₀ = (ℤ_≥‘0)
41	39, 40	eleqtrdi 2846	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
42		eluzfz1 13447	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘0) → 0 ∈ (0...𝑁))
43	41, 42	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ (0...𝑁))
44	43	anim1i 615	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (0 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑁)))
45		leisorel 14383	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (0 ≤ 𝑗 ↔ (𝑆‘0) ≤ (𝑆‘𝑗)))
47	20, 46	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁)) → (𝑆‘0) ≤ (𝑆‘𝑗))
48	47	3adant3 1132	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆‘𝑗)) → (𝑆‘0) ≤ (𝑆‘𝑗))
49		eqcom 2743	. . . . . . . 8 ⊢ (𝐴 = (𝑆‘𝑗) ↔ (𝑆‘𝑗) = 𝐴)
50	49	biimpi 216	. . . . . . 7 ⊢ (𝐴 = (𝑆‘𝑗) → (𝑆‘𝑗) = 𝐴)
51	50	3ad2ant3 1135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆‘𝑗)) → (𝑆‘𝑗) = 𝐴)
52	48, 51	breqtrd 5124	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆‘𝑗)) → (𝑆‘0) ≤ 𝐴)
53	52	rexlimdv3a 3141	. . . 4 ⊢ (𝜑 → (∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆‘𝑗) → (𝑆‘0) ≤ 𝐴))
54	18, 53	mpd 15	. . 3 ⊢ (𝜑 → (𝑆‘0) ≤ 𝐴)
55	3	rexrd 11182	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
56	4	rexrd 11182	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
57	13, 43	ffvelcdmd 7030	. . . 4 ⊢ (𝜑 → (𝑆‘0) ∈ (𝐴[,]𝐵))
58		iccgelb 13318	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑆‘0) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑆‘0))
59	55, 56, 57, 58	syl3anc 1373	. . 3 ⊢ (𝜑 → 𝐴 ≤ (𝑆‘0))
60	5, 57	sseldd 3934	. . . 4 ⊢ (𝜑 → (𝑆‘0) ∈ ℝ)
61	60, 3	letri3d 11275	. . 3 ⊢ (𝜑 → ((𝑆‘0) = 𝐴 ↔ ((𝑆‘0) ≤ 𝐴 ∧ 𝐴 ≤ (𝑆‘0))))
62	54, 59, 61	mpbir2and 713	. 2 ⊢ (𝜑 → (𝑆‘0) = 𝐴)
63		eluzfz2 13448	. . . . . 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 13317	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝑆‘𝑁) ∈ (𝐴[,]𝐵)) → (𝑆‘𝑁) ≤ 𝐵)
67	55, 56, 65, 66	syl3anc 1373	. . 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 584	. . . 4 ⊢ (𝜑 → ∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆‘𝑗))
71		simp3 1138	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝐵 = (𝑆‘𝑗))
72		elfzle2 13444	. . . . . . . 8 ⊢ (𝑗 ∈ (0...𝑁) → 𝑗 ≤ 𝑁)
73	72	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝑗 ≤ 𝑁)
74	9	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝑆 Isom < , < ((0...𝑁), 𝑇))
75	29	3adant3 1132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → ((0...𝑁) ⊆ ℝ^* ∧ 𝑇 ⊆ ℝ^*))
76		simp2 1137	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝑗 ∈ (0...𝑁))
77	64	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝑁 ∈ (0...𝑁))
78		leisorel 14383	. . . . . . . 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 1376	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → (𝑗 ≤ 𝑁 ↔ (𝑆‘𝑗) ≤ (𝑆‘𝑁)))
80	73, 79	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → (𝑆‘𝑗) ≤ (𝑆‘𝑁))
81	71, 80	eqbrtrd 5120	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆‘𝑗)) → 𝐵 ≤ (𝑆‘𝑁))
82	81	rexlimdv3a 3141	. . . 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 3934	. . . 4 ⊢ (𝜑 → (𝑆‘𝑁) ∈ ℝ)
85	84, 4	letri3d 11275	. . 3 ⊢ (𝜑 → ((𝑆‘𝑁) = 𝐵 ↔ ((𝑆‘𝑁) ≤ 𝐵 ∧ 𝐵 ≤ (𝑆‘𝑁))))
86	67, 83, 85	mpbir2and 713	. 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∃wrex 3060 ⊆ wss 3901 ∅c0 4285 class class class wbr 5098 ℩cio 6446 ⟶wf 6488 –onto→wfo 6490 –1-1-onto→wf1o 6491 ‘cfv 6492 Isom wiso 6493 (class class class)co 7358 Fincfn 8883 ℝcr 11025 0cc0 11026 1c1 11027 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167 − cmin 11364 ℕcn 12145 ℕ₀cn0 12401 ℤcz 12488 ℤ_≥cuz 12751 [,]cicc 13264 ...cfz 13423 ♯chash 14253

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 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-n0 12402 df-z 12489 df-uz 12752 df-icc 13268 df-fz 13424 df-hash 14254

This theorem is referenced by: fourierdlem103 46453 fourierdlem104 46454

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