Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fourierdlem34 Structured version   Visualization version   GIF version

Theorem fourierdlem34 46161
Description: A partition is one to one. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem34.p 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
fourierdlem34.m (𝜑𝑀 ∈ ℕ)
fourierdlem34.q (𝜑𝑄 ∈ (𝑃𝑀))
Assertion
Ref Expression
fourierdlem34 (𝜑𝑄:(0...𝑀)–1-1→ℝ)
Distinct variable groups:   𝐴,𝑚,𝑝   𝐵,𝑚,𝑝   𝑖,𝑀,𝑚,𝑝   𝑄,𝑖,𝑝   𝜑,𝑖
Allowed substitution hints:   𝜑(𝑚,𝑝)   𝐴(𝑖)   𝐵(𝑖)   𝑃(𝑖,𝑚,𝑝)   𝑄(𝑚)

Proof of Theorem fourierdlem34
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fourierdlem34.q . . . . 5 (𝜑𝑄 ∈ (𝑃𝑀))
2 fourierdlem34.m . . . . . 6 (𝜑𝑀 ∈ ℕ)
3 fourierdlem34.p . . . . . . 7 𝑃 = (𝑚 ∈ ℕ ↦ {𝑝 ∈ (ℝ ↑m (0...𝑚)) ∣ (((𝑝‘0) = 𝐴 ∧ (𝑝𝑚) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑚)(𝑝𝑖) < (𝑝‘(𝑖 + 1)))})
43fourierdlem2 46129 . . . . . 6 (𝑀 ∈ ℕ → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
52, 4syl 17 . . . . 5 (𝜑 → (𝑄 ∈ (𝑃𝑀) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1))))))
61, 5mpbid 232 . . . 4 (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
76simpld 494 . . 3 (𝜑𝑄 ∈ (ℝ ↑m (0...𝑀)))
8 elmapi 8890 . . 3 (𝑄 ∈ (ℝ ↑m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
97, 8syl 17 . 2 (𝜑𝑄:(0...𝑀)⟶ℝ)
10 simplr 768 . . . . . 6 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄𝑖) = (𝑄𝑗)) ∧ ¬ 𝑖 = 𝑗) → (𝑄𝑖) = (𝑄𝑗))
119ffvelcdmda 7103 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ ℝ)
1211ad2antrr 726 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄𝑖) ∈ ℝ)
139ffvelcdmda 7103 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (0...𝑀)) → (𝑄𝑘) ∈ ℝ)
1413ad4ant14 752 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0...𝑀)) → (𝑄𝑘) ∈ ℝ)
1514adantllr 719 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0...𝑀)) → (𝑄𝑘) ∈ ℝ)
16 eleq1w 2823 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑘 → (𝑖 ∈ (0..^𝑀) ↔ 𝑘 ∈ (0..^𝑀)))
1716anbi2d 630 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑘 → ((𝜑𝑖 ∈ (0..^𝑀)) ↔ (𝜑𝑘 ∈ (0..^𝑀))))
18 fveq2 6905 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑘 → (𝑄𝑖) = (𝑄𝑘))
19 oveq1 7439 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝑘 → (𝑖 + 1) = (𝑘 + 1))
2019fveq2d 6909 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑘 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑘 + 1)))
2118, 20breq12d 5155 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑘 → ((𝑄𝑖) < (𝑄‘(𝑖 + 1)) ↔ (𝑄𝑘) < (𝑄‘(𝑘 + 1))))
2217, 21imbi12d 344 . . . . . . . . . . . . . . 15 (𝑖 = 𝑘 → (((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1))) ↔ ((𝜑𝑘 ∈ (0..^𝑀)) → (𝑄𝑘) < (𝑄‘(𝑘 + 1)))))
236simprrd 773 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
2423r19.21bi 3250 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
2522, 24chvarvv 1997 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (0..^𝑀)) → (𝑄𝑘) < (𝑄‘(𝑘 + 1)))
2625ad4ant14 752 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑄𝑘) < (𝑄‘(𝑘 + 1)))
2726adantllr 719 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑄𝑘) < (𝑄‘(𝑘 + 1)))
28 simpllr 775 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 ∈ (0...𝑀))
29 simplr 768 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ (0...𝑀))
30 simpr 484 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
3115, 27, 28, 29, 30monoords 45314 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄𝑖) < (𝑄𝑗))
3212, 31ltned 11398 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄𝑖) ≠ (𝑄𝑗))
3332neneqd 2944 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → ¬ (𝑄𝑖) = (𝑄𝑗))
3433adantlr 715 . . . . . . . 8 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ 𝑖 < 𝑗) → ¬ (𝑄𝑖) = (𝑄𝑗))
35 simpll 766 . . . . . . . . 9 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → ((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)))
36 elfzelz 13565 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
3736zred 12724 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
3837ad3antlr 731 . . . . . . . . . 10 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ∈ ℝ)
39 elfzelz 13565 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
4039zred 12724 . . . . . . . . . . 11 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
4140ad4antlr 733 . . . . . . . . . 10 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑖 ∈ ℝ)
42 neqne 2947 . . . . . . . . . . . 12 𝑖 = 𝑗𝑖𝑗)
4342necomd 2995 . . . . . . . . . . 11 𝑖 = 𝑗𝑗𝑖)
4443ad2antlr 727 . . . . . . . . . 10 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑗𝑖)
45 simpr 484 . . . . . . . . . 10 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → ¬ 𝑖 < 𝑗)
4638, 41, 44, 45lttri5d 45316 . . . . . . . . 9 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑗 < 𝑖)
479ffvelcdmda 7103 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑄𝑗) ∈ ℝ)
4847adantr 480 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄𝑗) ∈ ℝ)
4948adantllr 719 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄𝑗) ∈ ℝ)
50 simp-4l 782 . . . . . . . . . . . . 13 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) ∧ 𝑘 ∈ (0...𝑀)) → 𝜑)
5150, 13sylancom 588 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) ∧ 𝑘 ∈ (0...𝑀)) → (𝑄𝑘) ∈ ℝ)
52 simp-4l 782 . . . . . . . . . . . . 13 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) ∧ 𝑘 ∈ (0..^𝑀)) → 𝜑)
5352, 25sylancom 588 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑄𝑘) < (𝑄‘(𝑘 + 1)))
54 simplr 768 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → 𝑗 ∈ (0...𝑀))
55 simpllr 775 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → 𝑖 ∈ (0...𝑀))
56 simpr 484 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → 𝑗 < 𝑖)
5751, 53, 54, 55, 56monoords 45314 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄𝑗) < (𝑄𝑖))
5849, 57gtned 11397 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄𝑖) ≠ (𝑄𝑗))
5958neneqd 2944 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → ¬ (𝑄𝑖) = (𝑄𝑗))
6035, 46, 59syl2anc 584 . . . . . . . 8 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → ¬ (𝑄𝑖) = (𝑄𝑗))
6134, 60pm2.61dan 812 . . . . . . 7 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) → ¬ (𝑄𝑖) = (𝑄𝑗))
6261adantlr 715 . . . . . 6 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄𝑖) = (𝑄𝑗)) ∧ ¬ 𝑖 = 𝑗) → ¬ (𝑄𝑖) = (𝑄𝑗))
6310, 62condan 817 . . . . 5 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄𝑖) = (𝑄𝑗)) → 𝑖 = 𝑗)
6463ex 412 . . . 4 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑄𝑖) = (𝑄𝑗) → 𝑖 = 𝑗))
6564ralrimiva 3145 . . 3 ((𝜑𝑖 ∈ (0...𝑀)) → ∀𝑗 ∈ (0...𝑀)((𝑄𝑖) = (𝑄𝑗) → 𝑖 = 𝑗))
6665ralrimiva 3145 . 2 (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)((𝑄𝑖) = (𝑄𝑗) → 𝑖 = 𝑗))
67 dff13 7276 . 2 (𝑄:(0...𝑀)–1-1→ℝ ↔ (𝑄:(0...𝑀)⟶ℝ ∧ ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)((𝑄𝑖) = (𝑄𝑗) → 𝑖 = 𝑗)))
689, 66, 67sylanbrc 583 1 (𝜑𝑄:(0...𝑀)–1-1→ℝ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wne 2939  wral 3060  {crab 3435   class class class wbr 5142  cmpt 5224  wf 6556  1-1wf1 6557  cfv 6560  (class class class)co 7432  m cmap 8867  cr 11155  0cc0 11156  1c1 11157   + caddc 11159   < clt 11296  cn 12267  ...cfz 13548  ..^cfzo 13695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-n0 12529  df-z 12616  df-uz 12880  df-fz 13549  df-fzo 13696
This theorem is referenced by:  fourierdlem50  46176
  Copyright terms: Public domain W3C validator