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Theorem fourierdlem34 44372
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 44340 . . . . . 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 8787 . . 3 (𝑄 ∈ (ℝ ↑m (0...𝑀)) → 𝑄:(0...𝑀)⟶ℝ)
97, 8syl 17 . 2 (𝜑𝑄:(0...𝑀)⟶ℝ)
10 simplr 767 . . . . . 6 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄𝑖) = (𝑄𝑗)) ∧ ¬ 𝑖 = 𝑗) → (𝑄𝑖) = (𝑄𝑗))
119ffvelcdmda 7035 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (0...𝑀)) → (𝑄𝑖) ∈ ℝ)
1211ad2antrr 724 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄𝑖) ∈ ℝ)
139ffvelcdmda 7035 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (0...𝑀)) → (𝑄𝑘) ∈ ℝ)
1413ad4ant14 750 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0...𝑀)) → (𝑄𝑘) ∈ ℝ)
1514adantllr 717 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0...𝑀)) → (𝑄𝑘) ∈ ℝ)
16 eleq1w 2820 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑘 → (𝑖 ∈ (0..^𝑀) ↔ 𝑘 ∈ (0..^𝑀)))
1716anbi2d 629 . . . . . . . . . . . . . . . 16 (𝑖 = 𝑘 → ((𝜑𝑖 ∈ (0..^𝑀)) ↔ (𝜑𝑘 ∈ (0..^𝑀))))
18 fveq2 6842 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑘 → (𝑄𝑖) = (𝑄𝑘))
19 oveq1 7364 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝑘 → (𝑖 + 1) = (𝑘 + 1))
2019fveq2d 6846 . . . . . . . . . . . . . . . . 17 (𝑖 = 𝑘 → (𝑄‘(𝑖 + 1)) = (𝑄‘(𝑘 + 1)))
2118, 20breq12d 5118 . . . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑖 ∈ (0..^𝑀)(𝑄𝑖) < (𝑄‘(𝑖 + 1)))
2423r19.21bi 3234 . . . . . . . . . . . . . . 15 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑄𝑖) < (𝑄‘(𝑖 + 1)))
2522, 24chvarvv 2002 . . . . . . . . . . . . . 14 ((𝜑𝑘 ∈ (0..^𝑀)) → (𝑄𝑘) < (𝑄‘(𝑘 + 1)))
2625ad4ant14 750 . . . . . . . . . . . . 13 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑄𝑘) < (𝑄‘(𝑘 + 1)))
2726adantllr 717 . . . . . . . . . . . 12 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑄𝑘) < (𝑄‘(𝑘 + 1)))
28 simpllr 774 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 ∈ (0...𝑀))
29 simplr 767 . . . . . . . . . . . 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 43521 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄𝑖) < (𝑄𝑗))
3212, 31ltned 11291 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → (𝑄𝑖) ≠ (𝑄𝑗))
3332neneqd 2948 . . . . . . . . 9 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑖 < 𝑗) → ¬ (𝑄𝑖) = (𝑄𝑗))
3433adantlr 713 . . . . . . . 8 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ 𝑖 < 𝑗) → ¬ (𝑄𝑖) = (𝑄𝑗))
35 simpll 765 . . . . . . . . 9 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → ((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)))
36 elfzelz 13441 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℤ)
3736zred 12607 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑀) → 𝑗 ∈ ℝ)
3837ad3antlr 729 . . . . . . . . . 10 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑗 ∈ ℝ)
39 elfzelz 13441 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℤ)
4039zred 12607 . . . . . . . . . . 11 (𝑖 ∈ (0...𝑀) → 𝑖 ∈ ℝ)
4140ad4antlr 731 . . . . . . . . . 10 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑖 ∈ ℝ)
42 neqne 2951 . . . . . . . . . . . 12 𝑖 = 𝑗𝑖𝑗)
4342necomd 2999 . . . . . . . . . . 11 𝑖 = 𝑗𝑗𝑖)
4443ad2antlr 725 . . . . . . . . . 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 43523 . . . . . . . . 9 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) ∧ ¬ 𝑖 < 𝑗) → 𝑗 < 𝑖)
479ffvelcdmda 7035 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0...𝑀)) → (𝑄𝑗) ∈ ℝ)
4847adantr 481 . . . . . . . . . . . 12 (((𝜑𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄𝑗) ∈ ℝ)
4948adantllr 717 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄𝑗) ∈ ℝ)
50 simp-4l 781 . . . . . . . . . . . . 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 781 . . . . . . . . . . . . 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 767 . . . . . . . . . . . 12 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → 𝑗 ∈ (0...𝑀))
55 simpllr 774 . . . . . . . . . . . 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 43521 . . . . . . . . . . 11 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄𝑗) < (𝑄𝑖))
5849, 57gtned 11290 . . . . . . . . . 10 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ 𝑗 < 𝑖) → (𝑄𝑖) ≠ (𝑄𝑗))
5958neneqd 2948 . . . . . . . . 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 811 . . . . . . 7 ((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ ¬ 𝑖 = 𝑗) → ¬ (𝑄𝑖) = (𝑄𝑗))
6261adantlr 713 . . . . . 6 (((((𝜑𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑀)) ∧ (𝑄𝑖) = (𝑄𝑗)) ∧ ¬ 𝑖 = 𝑗) → ¬ (𝑄𝑖) = (𝑄𝑗))
6310, 62condan 816 . . . . 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 3143 . . 3 ((𝜑𝑖 ∈ (0...𝑀)) → ∀𝑗 ∈ (0...𝑀)((𝑄𝑖) = (𝑄𝑗) → 𝑖 = 𝑗))
6665ralrimiva 3143 . 2 (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)((𝑄𝑖) = (𝑄𝑗) → 𝑖 = 𝑗))
67 dff13 7202 . 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 1541  wcel 2106  wne 2943  wral 3064  {crab 3407   class class class wbr 5105  cmpt 5188  wf 6492  1-1wf1 6493  cfv 6496  (class class class)co 7357  m cmap 8765  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189  cn 12153  ...cfz 13424  ..^cfzo 13567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568
This theorem is referenced by:  fourierdlem50  44387
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