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Theorem fourierdlem52 40795
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 40147 . . . . . 6 (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
62, 5sstrd 3719 . . . . 5 (𝜑𝑇 ⊆ ℝ)
7 fourierdlem52.s . . . . 5 𝑆 = (℩𝑓𝑓 Isom < , < ((0...𝑁), 𝑇))
8 fourierdlem52.n . . . . 5 𝑁 = ((♯‘𝑇) − 1)
91, 6, 7, 8fourierdlem36 40780 . . . 4 (𝜑𝑆 Isom < , < ((0...𝑁), 𝑇))
10 isof1o 6688 . . . 4 (𝑆 Isom < , < ((0...𝑁), 𝑇) → 𝑆:(0...𝑁)–1-1-onto𝑇)
11 f1of 6250 . . . 4 (𝑆:(0...𝑁)–1-1-onto𝑇𝑆:(0...𝑁)⟶𝑇)
129, 10, 113syl 18 . . 3 (𝜑𝑆:(0...𝑁)⟶𝑇)
1312, 2fssd 6170 . 2 (𝜑𝑆:(0...𝑁)⟶(𝐴[,]𝐵))
14 f1ofo 6257 . . . . . 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 6493 . . . . 5 ((𝑆:(0...𝑁)–onto𝑇𝐴𝑇) → ∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆𝑗))
1815, 16, 17syl2anc 696 . . . 4 (𝜑 → ∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆𝑗))
19 elfzle1 12458 . . . . . . . . 9 (𝑗 ∈ (0...𝑁) → 0 ≤ 𝑗)
2019adantl 473 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁)) → 0 ≤ 𝑗)
219adantr 472 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑆 Isom < , < ((0...𝑁), 𝑇))
22 ressxr 10196 . . . . . . . . . . . 12 ℝ ⊆ ℝ*
236, 22syl6ss 3721 . . . . . . . . . . 11 (𝜑𝑇 ⊆ ℝ*)
2423adantr 472 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0...𝑁)) → 𝑇 ⊆ ℝ*)
25 fzssz 12457 . . . . . . . . . . 11 (0...𝑁) ⊆ ℤ
26 zssre 11497 . . . . . . . . . . . 12 ℤ ⊆ ℝ
2726, 22sstri 3718 . . . . . . . . . . 11 ℤ ⊆ ℝ*
2825, 27sstri 3718 . . . . . . . . . 10 (0...𝑁) ⊆ ℝ*
2924, 28jctil 561 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → ((0...𝑁) ⊆ ℝ*𝑇 ⊆ ℝ*))
30 hashcl 13260 . . . . . . . . . . . . . . . 16 (𝑇 ∈ Fin → (♯‘𝑇) ∈ ℕ0)
311, 30syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (♯‘𝑇) ∈ ℕ0)
32 ne0i 4029 . . . . . . . . . . . . . . . . 17 (𝐴𝑇𝑇 ≠ ∅)
3316, 32syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑇 ≠ ∅)
34 hashge1 13291 . . . . . . . . . . . . . . . 16 ((𝑇 ∈ Fin ∧ 𝑇 ≠ ∅) → 1 ≤ (♯‘𝑇))
351, 33, 34syl2anc 696 . . . . . . . . . . . . . . 15 (𝜑 → 1 ≤ (♯‘𝑇))
36 elnnnn0c 11451 . . . . . . . . . . . . . . 15 ((♯‘𝑇) ∈ ℕ ↔ ((♯‘𝑇) ∈ ℕ0 ∧ 1 ≤ (♯‘𝑇)))
3731, 35, 36sylanbrc 701 . . . . . . . . . . . . . 14 (𝜑 → (♯‘𝑇) ∈ ℕ)
38 nnm1nn0 11447 . . . . . . . . . . . . . 14 ((♯‘𝑇) ∈ ℕ → ((♯‘𝑇) − 1) ∈ ℕ0)
3937, 38syl 17 . . . . . . . . . . . . 13 (𝜑 → ((♯‘𝑇) − 1) ∈ ℕ0)
408, 39syl5eqel 2807 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℕ0)
41 nn0uz 11836 . . . . . . . . . . . 12 0 = (ℤ‘0)
4240, 41syl6eleq 2813 . . . . . . . . . . 11 (𝜑𝑁 ∈ (ℤ‘0))
43 eluzfz1 12462 . . . . . . . . . . 11 (𝑁 ∈ (ℤ‘0) → 0 ∈ (0...𝑁))
4442, 43syl 17 . . . . . . . . . 10 (𝜑 → 0 ∈ (0...𝑁))
4544anim1i 593 . . . . . . . . 9 ((𝜑𝑗 ∈ (0...𝑁)) → (0 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...𝑁)))
46 leisorel 13357 . . . . . . . . 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 1439 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁)) → (0 ≤ 𝑗 ↔ (𝑆‘0) ≤ (𝑆𝑗)))
4820, 47mpbid 222 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁)) → (𝑆‘0) ≤ (𝑆𝑗))
49483adant3 1124 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆𝑗)) → (𝑆‘0) ≤ (𝑆𝑗))
50 eqcom 2731 . . . . . . . 8 (𝐴 = (𝑆𝑗) ↔ (𝑆𝑗) = 𝐴)
5150biimpi 206 . . . . . . 7 (𝐴 = (𝑆𝑗) → (𝑆𝑗) = 𝐴)
52513ad2ant3 1127 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆𝑗)) → (𝑆𝑗) = 𝐴)
5349, 52breqtrd 4786 . . . . 5 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐴 = (𝑆𝑗)) → (𝑆‘0) ≤ 𝐴)
5453rexlimdv3a 3135 . . . 4 (𝜑 → (∃𝑗 ∈ (0...𝑁)𝐴 = (𝑆𝑗) → (𝑆‘0) ≤ 𝐴))
5518, 54mpd 15 . . 3 (𝜑 → (𝑆‘0) ≤ 𝐴)
563rexrd 10202 . . . 4 (𝜑𝐴 ∈ ℝ*)
574rexrd 10202 . . . 4 (𝜑𝐵 ∈ ℝ*)
5813, 44ffvelrnd 6475 . . . 4 (𝜑 → (𝑆‘0) ∈ (𝐴[,]𝐵))
59 iccgelb 12344 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (𝑆‘0) ∈ (𝐴[,]𝐵)) → 𝐴 ≤ (𝑆‘0))
6056, 57, 58, 59syl3anc 1439 . . 3 (𝜑𝐴 ≤ (𝑆‘0))
615, 58sseldd 3710 . . . 4 (𝜑 → (𝑆‘0) ∈ ℝ)
6261, 3letri3d 10292 . . 3 (𝜑 → ((𝑆‘0) = 𝐴 ↔ ((𝑆‘0) ≤ 𝐴𝐴 ≤ (𝑆‘0))))
6355, 60, 62mpbir2and 995 . 2 (𝜑 → (𝑆‘0) = 𝐴)
64 eluzfz2 12463 . . . . . 6 (𝑁 ∈ (ℤ‘0) → 𝑁 ∈ (0...𝑁))
6542, 64syl 17 . . . . 5 (𝜑𝑁 ∈ (0...𝑁))
6613, 65ffvelrnd 6475 . . . 4 (𝜑 → (𝑆𝑁) ∈ (𝐴[,]𝐵))
67 iccleub 12343 . . . 4 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ* ∧ (𝑆𝑁) ∈ (𝐴[,]𝐵)) → (𝑆𝑁) ≤ 𝐵)
6856, 57, 66, 67syl3anc 1439 . . 3 (𝜑 → (𝑆𝑁) ≤ 𝐵)
69 fourierdlem52.bt . . . . 5 (𝜑𝐵𝑇)
70 foelrn 6493 . . . . 5 ((𝑆:(0...𝑁)–onto𝑇𝐵𝑇) → ∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆𝑗))
7115, 69, 70syl2anc 696 . . . 4 (𝜑 → ∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆𝑗))
72 simp3 1130 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝐵 = (𝑆𝑗))
73 elfzle2 12459 . . . . . . . 8 (𝑗 ∈ (0...𝑁) → 𝑗𝑁)
74733ad2ant2 1126 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝑗𝑁)
7593ad2ant1 1125 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝑆 Isom < , < ((0...𝑁), 𝑇))
76293adant3 1124 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → ((0...𝑁) ⊆ ℝ*𝑇 ⊆ ℝ*))
77 simp2 1129 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝑗 ∈ (0...𝑁))
78653ad2ant1 1125 . . . . . . . 8 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝑁 ∈ (0...𝑁))
79 leisorel 13357 . . . . . . . 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 1443 . . . . . . 7 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → (𝑗𝑁 ↔ (𝑆𝑗) ≤ (𝑆𝑁)))
8174, 80mpbid 222 . . . . . 6 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → (𝑆𝑗) ≤ (𝑆𝑁))
8272, 81eqbrtrd 4782 . . . . 5 ((𝜑𝑗 ∈ (0...𝑁) ∧ 𝐵 = (𝑆𝑗)) → 𝐵 ≤ (𝑆𝑁))
8382rexlimdv3a 3135 . . . 4 (𝜑 → (∃𝑗 ∈ (0...𝑁)𝐵 = (𝑆𝑗) → 𝐵 ≤ (𝑆𝑁)))
8471, 83mpd 15 . . 3 (𝜑𝐵 ≤ (𝑆𝑁))
855, 66sseldd 3710 . . . 4 (𝜑 → (𝑆𝑁) ∈ ℝ)
8685, 4letri3d 10292 . . 3 (𝜑 → ((𝑆𝑁) = 𝐵 ↔ ((𝑆𝑁) ≤ 𝐵𝐵 ≤ (𝑆𝑁))))
8768, 84, 86mpbir2and 995 . 2 (𝜑 → (𝑆𝑁) = 𝐵)
8813, 63, 87jca31 558 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 383  w3a 1072   = wceq 1596  wcel 2103  wne 2896  wrex 3015  wss 3680  c0 4023   class class class wbr 4760  cio 5962  wf 5997  ontowfo 5999  1-1-ontowf1o 6000  cfv 6001   Isom wiso 6002  (class class class)co 6765  Fincfn 8072  cr 10048  0cc0 10049  1c1 10050  *cxr 10186   < clt 10187  cle 10188  cmin 10379  cn 11133  0cn0 11405  cz 11490  cuz 11800  [,]cicc 12292  ...cfz 12440  chash 13232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-inf2 8651  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-se 5178  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-isom 6010  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-oi 8531  df-card 8878  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-n0 11406  df-z 11491  df-uz 11801  df-icc 12296  df-fz 12441  df-hash 13233
This theorem is referenced by:  fourierdlem103  40846  fourierdlem104  40847
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