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Theorem fourierdlem34 44007
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 43975 . . . . . 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 231 . . . 4 (𝜑 → (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((𝑄‘0) = 𝐴 ∧ (𝑄𝑀) = 𝐵) ∧ ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))))
76simpld 495 . . 3 (𝜑𝑄 ∈ (ℝ ↑m (0...𝑀)))
8 elmapi 8700 . . 3 (𝑄 ∈ (ℝ ↑m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
97, 8syl 17 . 2 (𝜑𝑄:(0...𝑀)⟶ℝ)
10 simplr 766 . . . . . 6 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄𝑖) = (𝑄𝑗)) ∧ ¬ 𝑖 = 𝑗) → (𝑄𝑖) = (𝑄𝑗))
119ffvelcdmda 7011 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ ℝ)
1211ad2antrr 723 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄𝑖) ∈ ℝ)
139ffvelcdmda 7011 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (0...𝑀)) → (𝑄𝑘) ∈ ℝ)
1413ad4ant14 749 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0...𝑀)) → (𝑄𝑘) ∈ ℝ)
1514adantllr 716 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0...𝑀)) → (𝑄𝑘) ∈ ℝ)
16 eleq1w 2819 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑘 → (𝑖 ∈ (0..^𝑀) ↔ 𝑘 ∈ (0..^𝑀)))
1716anbi2d 629 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑘 → ((𝜑𝑖 ∈ (0..^𝑀)) ↔ (𝜑𝑘 ∈ (0..^𝑀))))
18 fveq2 6819 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑘 → (𝑄𝑖) = (𝑄𝑘))
19 oveq1 7336 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝑘 → (𝑖 + 1) = (𝑘 + 1))
2019fveq2d 6823 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑘 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑘 + 1)))
2118, 20breq12d 5102 . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
2423r19.21bi 3230 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
2522, 24chvarvv 2001 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (0..^𝑀)) → (𝑄𝑘) < (𝑄‘(𝑘 + 1)))
2625ad4ant14 749 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑄𝑘) < (𝑄‘(𝑘 + 1)))
2726adantllr 716 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑄𝑘) < (𝑄‘(𝑘 + 1)))
28 simpllr 773 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 ∈ (0...𝑀))
29 simplr 766 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ (0...𝑀))
30 simpr 485 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
3115, 27, 28, 29, 30monoords 43160 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄𝑖) < (𝑄𝑗))
3212, 31ltned 11204 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄𝑖) ≠ (𝑄𝑗))
3332neneqd 2945 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → ¬ (𝑄𝑖) = (𝑄𝑗))
3433adantlr 712 . . . . . . . 8 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ 𝑖 < 𝑗) → ¬ (𝑄𝑖) = (𝑄𝑗))
35 simpll 764 . . . . . . . . 9 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → ((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)))
36 elfzelz 13349 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
3736zred 12519 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
3837ad3antlr 728 . . . . . . . . . 10 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ∈ ℝ)
39 elfzelz 13349 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
4039zred 12519 . . . . . . . . . . 11 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
4140ad4antlr 730 . . . . . . . . . 10 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑖 ∈ ℝ)
42 neqne 2948 . . . . . . . . . . . 12 𝑖 = 𝑗𝑖𝑗)
4342necomd 2996 . . . . . . . . . . 11 𝑖 = 𝑗𝑗𝑖)
4443ad2antlr 724 . . . . . . . . . 10 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑗𝑖)
45 simpr 485 . . . . . . . . . 10 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → ¬ 𝑖 < 𝑗)
4638, 41, 44, 45lttri5d 43162 . . . . . . . . 9 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑗 < 𝑖)
479ffvelcdmda 7011 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑄𝑗) ∈ ℝ)
4847adantr 481 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄𝑗) ∈ ℝ)
4948adantllr 716 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄𝑗) ∈ ℝ)
50 simp-4l 780 . . . . . . . . . . . . 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 780 . . . . . . . . . . . . 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 766 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → 𝑗 ∈ (0...𝑀))
55 simpllr 773 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → 𝑖 ∈ (0...𝑀))
56 simpr 485 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → 𝑗 < 𝑖)
5751, 53, 54, 55, 56monoords 43160 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄𝑗) < (𝑄𝑖))
5849, 57gtned 11203 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄𝑖) ≠ (𝑄𝑗))
5958neneqd 2945 . . . . . . . . 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 810 . . . . . . 7 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) → ¬ (𝑄𝑖) = (𝑄𝑗))
6261adantlr 712 . . . . . 6 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄𝑖) = (𝑄𝑗)) ∧ ¬ 𝑖 = 𝑗) → ¬ (𝑄𝑖) = (𝑄𝑗))
6310, 62condan 815 . . . . 5 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄𝑖) = (𝑄𝑗)) → 𝑖 = 𝑗)
6463ex 413 . . . 4 (((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) → ((𝑄𝑖) = (𝑄𝑗) → 𝑖 = 𝑗))
6564ralrimiva 3139 . . 3 ((𝜑𝑖 ∈ (0...𝑀)) → ∀𝑗 ∈ (0...𝑀)((𝑄𝑖) = (𝑄𝑗) → 𝑖 = 𝑗))
6665ralrimiva 3139 . 2 (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)((𝑄𝑖) = (𝑄𝑗) → 𝑖 = 𝑗))
67 dff13 7178 . 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 205  wa 396   = wceq 1540  wcel 2105  wne 2940  wral 3061  {crab 3403   class class class wbr 5089  cmpt 5172  wf 6469  1-1wf1 6470  cfv 6473  (class class class)co 7329  m cmap 8678  cr 10963  0cc0 10964  1c1 10965   + caddc 10967   < clt 11102  cn 12066  ...cfz 13332  ..^cfzo 13475
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-om 7773  df-1st 7891  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-er 8561  df-map 8680  df-en 8797  df-dom 8798  df-sdom 8799  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-nn 12067  df-n0 12327  df-z 12413  df-uz 12676  df-fz 13333  df-fzo 13476
This theorem is referenced by:  fourierdlem50  44022
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