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Theorem fourierdlem52 39669
Description: d16:d17,d18:jca |- ( ph -> ( ( S 0) ≤ 𝐴𝐴 ≤ (𝑆 0 ) ) ) . (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem52.tf (𝜑𝑇 ∈ Fin)
fourierdlem52.n 𝑁 = ((#‘𝑇) − 1)
fourierdlem52.s 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))
fourierdlem52.a (𝜑𝐴 ∈ ℝ)
fourierdlem52.b (𝜑𝐵 ∈ ℝ)
fourierdlem52.t (𝜑𝑇 ⊆ (𝐴[,]𝐵))
fourierdlem52.at (𝜑𝐴𝑇)
fourierdlem52.bt (𝜑𝐵𝑇)
Assertion
Ref Expression
fourierdlem52 (𝜑 → ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (𝑆‘0) = 𝐴) ∧ (𝑆𝑁) = 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.
StepHypRef Expression
1 fourierdlem52.tf . . . . 5 (𝜑𝑇 ∈ Fin)
2 fourierdlem52.t . . . . . 6 (𝜑𝑇 ⊆ (𝐴[,]𝐵))
3 fourierdlem52.a . . . . . . 7 (𝜑𝐴 ∈ ℝ)
4 fourierdlem52.b . . . . . . 7 (𝜑𝐵 ∈ ℝ)
53, 4iccssred 39125 . . . . . 6 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
62, 5sstrd 3598 . . . . 5 (𝜑𝑇 ⊆ ℝ)
7 fourierdlem52.s . . . . 5 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))
8 fourierdlem52.n . . . . 5 𝑁 = ((#‘𝑇) − 1)
91, 6, 7, 8fourierdlem36 39654 . . . 4 (𝜑𝑆 Isom < , < ((0...𝑁), 𝑇))
10 isof1o 6528 . . . 4 (𝑆 Isom < , < ((0...𝑁), 𝑇) → 𝑆:(0...𝑁)–1-1-onto𝑇)
11 f1of 6096 . . . 4 (𝑆:(0...𝑁)–1-1-onto𝑇𝑆:(0...𝑁)⟶𝑇)
129, 10, 113syl 18 . . 3 (𝜑𝑆:(0...𝑁)⟶𝑇)
1312, 2fssd 6016 . 2 (𝜑𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
14 f1ofo 6103 . . . . . 6 (𝑆:(0...𝑁)–1-1-onto𝑇𝑆:(0...𝑁)–onto𝑇)
159, 10, 143syl 18 . . . . 5 (𝜑𝑆:(0...𝑁)–onto𝑇)
16 fourierdlem52.at . . . . 5 (𝜑𝐴𝑇)
17 foelrn 6335 . . . . 5 ((𝑆:(0...𝑁)–onto𝑇𝐴𝑇) → ∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆𝑗))
1815, 16, 17syl2anc 692 . . . 4 (𝜑 → ∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆𝑗))
19 elfzle1 12283 . . . . . . . . 9 (𝑗 ∈ (0...𝑁) → 0 ≤ 𝑗)
2019adantl 482 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁)) → 0 ≤ 𝑗)
219adantr 481 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑆 Isom < , < ((0...𝑁), 𝑇))
22 ressxr 10028 . . . . . . . . . . . 12 ℝ ⊆ ℝ*
236, 22syl6ss 3600 . . . . . . . . . . 11 (𝜑𝑇 ⊆ ℝ*)
2423adantr 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑇 ⊆ ℝ*)
25 fzssz 12282 . . . . . . . . . . 11 (0...𝑁) ⊆ ℤ
26 zssre 11329 . . . . . . . . . . . 12 ℤ ⊆ ℝ
2726, 22sstri 3597 . . . . . . . . . . 11 ℤ ⊆ ℝ*
2825, 27sstri 3597 . . . . . . . . . 10 (0...𝑁) ⊆ ℝ*
2924, 28jctil 559 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → ((0...𝑁) ⊆ ℝ*𝑇 ⊆ ℝ*))
30 hashcl 13084 . . . . . . . . . . . . . . . 16 (𝑇 ∈ Fin → (#‘𝑇) ∈ ℕ0)
311, 30syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (#‘𝑇) ∈ ℕ0)
32 ne0i 3902 . . . . . . . . . . . . . . . . 17 (𝐴𝑇𝑇 ≠ ∅)
3316, 32syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑇 ≠ ∅)
34 hashge1 13115 . . . . . . . . . . . . . . . 16 ((𝑇 ∈ Fin ∧ 𝑇 ≠ ∅) → 1 ≤ (#‘𝑇))
351, 33, 34syl2anc 692 . . . . . . . . . . . . . . 15 (𝜑 → 1 ≤ (#‘𝑇))
36 elnnnn0c 11283 . . . . . . . . . . . . . . 15 ((#‘𝑇) ∈ ℕ ↔ ((#‘𝑇) ∈ ℕ0 ∧ 1 ≤ (#‘𝑇)))
3731, 35, 36sylanbrc 697 . . . . . . . . . . . . . 14 (𝜑 → (#‘𝑇) ∈ ℕ)
38 nnm1nn0 11279 . . . . . . . . . . . . . 14 ((#‘𝑇) ∈ ℕ → ((#‘𝑇) − 1) ∈ ℕ0)
3937, 38syl 17 . . . . . . . . . . . . 13 (𝜑 → ((#‘𝑇) − 1) ∈ ℕ0)
408, 39syl5eqel 2708 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
41 nn0uz 11666 . . . . . . . . . . . 12 0 = (ℤ‘0)
4240, 41syl6eleq 2714 . . . . . . . . . . 11 (𝜑𝑁 ∈ (ℤ‘0))
43 eluzfz1 12287 . . . . . . . . . . 11 (𝑁 ∈ (ℤ‘0) → 0 ∈ (0...𝑁))
4442, 43syl 17 . . . . . . . . . 10 (𝜑 → 0 ∈ (0...𝑁))
4544anim1i 591 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → (0 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑁)))
46 leisorel 13179 . . . . . . . . 9 ((𝑆 Isom < , < ((0...𝑁), 𝑇) ∧ ((0...𝑁) ⊆ ℝ*𝑇 ⊆ ℝ*) ∧ (0 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑁))) → (0 ≤ 𝑗 ↔ (𝑆‘0) ≤ (𝑆𝑗)))
4721, 29, 45, 46syl3anc 1323 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁)) → (0 ≤ 𝑗 ↔ (𝑆‘0) ≤ (𝑆𝑗)))
4820, 47mpbid 222 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑆‘0) ≤ (𝑆𝑗))
49483adant3 1079 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆𝑗)) → (𝑆‘0) ≤ (𝑆𝑗))
50 eqcom 2633 . . . . . . . 8 (𝐴 = (𝑆𝑗) ↔ (𝑆𝑗) = 𝐴)
5150biimpi 206 . . . . . . 7 (𝐴 = (𝑆𝑗) → (𝑆𝑗) = 𝐴)
52513ad2ant3 1082 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆𝑗)) → (𝑆𝑗) = 𝐴)
5349, 52breqtrd 4644 . . . . 5 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆𝑗)) → (𝑆‘0) ≤ 𝐴)
5453rexlimdv3a 3031 . . . 4 (𝜑 → (∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆𝑗) → (𝑆‘0) ≤ 𝐴))
5518, 54mpd 15 . . 3 (𝜑 → (𝑆‘0) ≤ 𝐴)
563rexrd 10034 . . . 4 (𝜑𝐴 ∈ ℝ*)
574rexrd 10034 . . . 4 (𝜑𝐵 ∈ ℝ*)
5813, 44ffvelrnd 6317 . . . 4 (𝜑 → (𝑆‘0) ∈ (𝐴[,]𝐵))
59 iccgelb 12169 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (𝑆‘0) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑆‘0))
6056, 57, 58, 59syl3anc 1323 . . 3 (𝜑𝐴 ≤ (𝑆‘0))
615, 58sseldd 3589 . . . 4 (𝜑 → (𝑆‘0) ∈ ℝ)
6261, 3letri3d 10124 . . 3 (𝜑 → ((𝑆‘0) = 𝐴 ↔ ((𝑆‘0) ≤ 𝐴𝐴 ≤ (𝑆‘0))))
6355, 60, 62mpbir2and 956 . 2 (𝜑 → (𝑆‘0) = 𝐴)
64 eluzfz2 12288 . . . . . 6 (𝑁 ∈ (ℤ‘0) → 𝑁 ∈ (0...𝑁))
6542, 64syl 17 . . . . 5 (𝜑𝑁 ∈ (0...𝑁))
6613, 65ffvelrnd 6317 . . . 4 (𝜑 → (𝑆𝑁) ∈ (𝐴[,]𝐵))
67 iccleub 12168 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (𝑆𝑁) ∈ (𝐴[,]𝐵)) → (𝑆𝑁) ≤ 𝐵)
6856, 57, 66, 67syl3anc 1323 . . 3 (𝜑 → (𝑆𝑁) ≤ 𝐵)
69 fourierdlem52.bt . . . . 5 (𝜑𝐵𝑇)
70 foelrn 6335 . . . . 5 ((𝑆:(0...𝑁)–onto𝑇𝐵𝑇) → ∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆𝑗))
7115, 69, 70syl2anc 692 . . . 4 (𝜑 → ∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆𝑗))
72 simp3 1061 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝐵 = (𝑆𝑗))
73 elfzle2 12284 . . . . . . . 8 (𝑗 ∈ (0...𝑁) → 𝑗𝑁)
74733ad2ant2 1081 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝑗𝑁)
7593ad2ant1 1080 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝑆 Isom < , < ((0...𝑁), 𝑇))
76293adant3 1079 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → ((0...𝑁) ⊆ ℝ*𝑇 ⊆ ℝ*))
77 simp2 1060 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝑗 ∈ (0...𝑁))
78653ad2ant1 1080 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝑁 ∈ (0...𝑁))
79 leisorel 13179 . . . . . . . 8 ((𝑆 Isom < , < ((0...𝑁), 𝑇) ∧ ((0...𝑁) ⊆ ℝ*𝑇 ⊆ ℝ*) ∧ (𝑗 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...𝑁))) → (𝑗𝑁 ↔ (𝑆𝑗) ≤ (𝑆𝑁)))
8075, 76, 77, 78, 79syl112anc 1327 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → (𝑗𝑁 ↔ (𝑆𝑗) ≤ (𝑆𝑁)))
8174, 80mpbid 222 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → (𝑆𝑗) ≤ (𝑆𝑁))
8272, 81eqbrtrd 4640 . . . . 5 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝐵 ≤ (𝑆𝑁))
8382rexlimdv3a 3031 . . . 4 (𝜑 → (∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆𝑗) → 𝐵 ≤ (𝑆𝑁)))
8471, 83mpd 15 . . 3 (𝜑𝐵 ≤ (𝑆𝑁))
855, 66sseldd 3589 . . . 4 (𝜑 → (𝑆𝑁) ∈ ℝ)
8685, 4letri3d 10124 . . 3 (𝜑 → ((𝑆𝑁) = 𝐵 ↔ ((𝑆𝑁) ≤ 𝐵𝐵 ≤ (𝑆𝑁))))
8768, 84, 86mpbir2and 956 . 2 (𝜑 → (𝑆𝑁) = 𝐵)
8813, 63, 87jca31 556 1 (𝜑 → ((𝑆:(0...𝑁)⟶(𝐴[,]𝐵) ∧ (𝑆‘0) = 𝐴) ∧ (𝑆𝑁) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wne 2796  wrex 2913  wss 3560  c0 3896   class class class wbr 4618  cio 5811  wf 5846  ontowfo 5848  1-1-ontowf1o 5849  cfv 5850   Isom wiso 5851  (class class class)co 6605  Fincfn 7900  cr 9880  0cc0 9881  1c1 9882  *cxr 10018   < clt 10019  cle 10020  cmin 10211  cn 10965  0cn0 11237  cz 11322  cuz 11631  [,]cicc 12117  ...cfz 12265  #chash 13054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-oi 8360  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-n0 11238  df-z 11323  df-uz 11632  df-icc 12121  df-fz 12266  df-hash 13055
This theorem is referenced by:  fourierdlem103  39720  fourierdlem104  39721
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