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Theorem fourierdlem15 45433
Description: The range of the partition is between its starting point and its ending point. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem15.1 𝑃 = (π‘š ∈ β„• ↦ {𝑝 ∈ (ℝ ↑m (0...π‘š)) ∣ (((π‘β€˜0) = 𝐴 ∧ (π‘β€˜π‘š) = 𝐡) ∧ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)))})
fourierdlem15.2 (πœ‘ β†’ 𝑀 ∈ β„•)
fourierdlem15.3 (πœ‘ β†’ 𝑄 ∈ (π‘ƒβ€˜π‘€))
Assertion
Ref Expression
fourierdlem15 (πœ‘ β†’ 𝑄:(0...𝑀)⟢(𝐴[,]𝐡))
Distinct variable groups:   𝐴,𝑖,π‘š,𝑝   𝐡,𝑖,π‘š,𝑝   𝑖,𝑀,π‘š,𝑝   𝑄,𝑖,𝑝   πœ‘,𝑖
Allowed substitution hints:   πœ‘(π‘š,𝑝)   𝑃(𝑖,π‘š,𝑝)   𝑄(π‘š)
Proof of Theorem fourierdlem15
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 fourierdlem15.3 . . . . . 6 (πœ‘ β†’ 𝑄 ∈ (π‘ƒβ€˜π‘€))
2 fourierdlem15.2 . . . . . . 7 (πœ‘ β†’ 𝑀 ∈ β„•)
3 fourierdlem15.1 . . . . . . . 8 𝑃 = (π‘š ∈ β„• ↦ {𝑝 ∈ (ℝ ↑m (0...π‘š)) ∣ (((π‘β€˜0) = 𝐴 ∧ (π‘β€˜π‘š) = 𝐡) ∧ βˆ€π‘– ∈ (0..^π‘š)(π‘β€˜π‘–) < (π‘β€˜(𝑖 + 1)))})
43fourierdlem2 45420 . . . . . . 7 (𝑀 ∈ β„• β†’ (𝑄 ∈ (π‘ƒβ€˜π‘€) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((π‘„β€˜0) = 𝐴 ∧ (π‘„β€˜π‘€) = 𝐡) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1))))))
52, 4syl 17 . . . . . 6 (πœ‘ β†’ (𝑄 ∈ (π‘ƒβ€˜π‘€) ↔ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((π‘„β€˜0) = 𝐴 ∧ (π‘„β€˜π‘€) = 𝐡) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1))))))
61, 5mpbid 231 . . . . 5 (πœ‘ β†’ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ∧ (((π‘„β€˜0) = 𝐴 ∧ (π‘„β€˜π‘€) = 𝐡) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1)))))
76simpld 494 . . . 4 (πœ‘ β†’ 𝑄 ∈ (ℝ ↑m (0...𝑀)))
8 reex 11221 . . . . . 6 ℝ ∈ V
98a1i 11 . . . . 5 (πœ‘ β†’ ℝ ∈ V)
10 ovex 7447 . . . . . 6 (0...𝑀) ∈ V
1110a1i 11 . . . . 5 (πœ‘ β†’ (0...𝑀) ∈ V)
129, 11elmapd 8850 . . . 4 (πœ‘ β†’ (𝑄 ∈ (ℝ ↑m (0...𝑀)) ↔ 𝑄:(0...𝑀)βŸΆβ„))
137, 12mpbid 231 . . 3 (πœ‘ β†’ 𝑄:(0...𝑀)βŸΆβ„)
14 ffn 6716 . . 3 (𝑄:(0...𝑀)βŸΆβ„ β†’ 𝑄 Fn (0...𝑀))
1513, 14syl 17 . 2 (πœ‘ β†’ 𝑄 Fn (0...𝑀))
166simprd 495 . . . . . . . . 9 (πœ‘ β†’ (((π‘„β€˜0) = 𝐴 ∧ (π‘„β€˜π‘€) = 𝐡) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1))))
1716simpld 494 . . . . . . . 8 (πœ‘ β†’ ((π‘„β€˜0) = 𝐴 ∧ (π‘„β€˜π‘€) = 𝐡))
1817simpld 494 . . . . . . 7 (πœ‘ β†’ (π‘„β€˜0) = 𝐴)
19 nnnn0 12501 . . . . . . . . . . 11 (𝑀 ∈ β„• β†’ 𝑀 ∈ β„•0)
20 nn0uz 12886 . . . . . . . . . . 11 β„•0 = (β„€β‰₯β€˜0)
2119, 20eleqtrdi 2838 . . . . . . . . . 10 (𝑀 ∈ β„• β†’ 𝑀 ∈ (β„€β‰₯β€˜0))
222, 21syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝑀 ∈ (β„€β‰₯β€˜0))
23 eluzfz1 13532 . . . . . . . . 9 (𝑀 ∈ (β„€β‰₯β€˜0) β†’ 0 ∈ (0...𝑀))
2422, 23syl 17 . . . . . . . 8 (πœ‘ β†’ 0 ∈ (0...𝑀))
2513, 24ffvelcdmd 7089 . . . . . . 7 (πœ‘ β†’ (π‘„β€˜0) ∈ ℝ)
2618, 25eqeltrrd 2829 . . . . . 6 (πœ‘ β†’ 𝐴 ∈ ℝ)
2726adantr 480 . . . . 5 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ 𝐴 ∈ ℝ)
2817simprd 495 . . . . . . 7 (πœ‘ β†’ (π‘„β€˜π‘€) = 𝐡)
29 eluzfz2 13533 . . . . . . . . 9 (𝑀 ∈ (β„€β‰₯β€˜0) β†’ 𝑀 ∈ (0...𝑀))
3022, 29syl 17 . . . . . . . 8 (πœ‘ β†’ 𝑀 ∈ (0...𝑀))
3113, 30ffvelcdmd 7089 . . . . . . 7 (πœ‘ β†’ (π‘„β€˜π‘€) ∈ ℝ)
3228, 31eqeltrrd 2829 . . . . . 6 (πœ‘ β†’ 𝐡 ∈ ℝ)
3332adantr 480 . . . . 5 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ 𝐡 ∈ ℝ)
3413ffvelcdmda 7088 . . . . 5 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ (π‘„β€˜π‘–) ∈ ℝ)
3518eqcomd 2733 . . . . . . 7 (πœ‘ β†’ 𝐴 = (π‘„β€˜0))
3635adantr 480 . . . . . 6 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ 𝐴 = (π‘„β€˜0))
37 elfzuz 13521 . . . . . . . 8 (𝑖 ∈ (0...𝑀) β†’ 𝑖 ∈ (β„€β‰₯β€˜0))
3837adantl 481 . . . . . . 7 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ 𝑖 ∈ (β„€β‰₯β€˜0))
3913ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑄:(0...𝑀)βŸΆβ„)
40 0zd 12592 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 0 ∈ β„€)
41 elfzel2 13523 . . . . . . . . . . 11 (𝑖 ∈ (0...𝑀) β†’ 𝑀 ∈ β„€)
4241adantr 480 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑀 ∈ β„€)
43 elfzelz 13525 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑖) β†’ 𝑗 ∈ β„€)
4443adantl 481 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑗 ∈ β„€)
45 elfzle1 13528 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑖) β†’ 0 ≀ 𝑗)
4645adantl 481 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 0 ≀ 𝑗)
4743zred 12688 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑖) β†’ 𝑗 ∈ ℝ)
4847adantl 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑗 ∈ ℝ)
49 elfzelz 13525 . . . . . . . . . . . . 13 (𝑖 ∈ (0...𝑀) β†’ 𝑖 ∈ β„€)
5049zred 12688 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) β†’ 𝑖 ∈ ℝ)
5150adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑖 ∈ ℝ)
5241zred 12688 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) β†’ 𝑀 ∈ ℝ)
5352adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑀 ∈ ℝ)
54 elfzle2 13529 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑖) β†’ 𝑗 ≀ 𝑖)
5554adantl 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑗 ≀ 𝑖)
56 elfzle2 13529 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) β†’ 𝑖 ≀ 𝑀)
5756adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑖 ≀ 𝑀)
5848, 51, 53, 55, 57letrd 11393 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑗 ≀ 𝑀)
5940, 42, 44, 46, 58elfzd 13516 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑗 ∈ (0...𝑀))
6059adantll 713 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑗 ∈ (0...𝑀))
6139, 60ffvelcdmd 7089 . . . . . . 7 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) β†’ (π‘„β€˜π‘—) ∈ ℝ)
62 simpll 766 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ πœ‘)
63 elfzle1 13528 . . . . . . . . . . 11 (𝑗 ∈ (0...(𝑖 βˆ’ 1)) β†’ 0 ≀ 𝑗)
6463adantl 481 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 0 ≀ 𝑗)
65 elfzelz 13525 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑖 βˆ’ 1)) β†’ 𝑗 ∈ β„€)
6665zred 12688 . . . . . . . . . . . 12 (𝑗 ∈ (0...(𝑖 βˆ’ 1)) β†’ 𝑗 ∈ ℝ)
6766adantl 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 ∈ ℝ)
6850adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑖 ∈ ℝ)
6952adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑀 ∈ ℝ)
70 peano2rem 11549 . . . . . . . . . . . . 13 (𝑖 ∈ ℝ β†’ (𝑖 βˆ’ 1) ∈ ℝ)
7168, 70syl 17 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ (𝑖 βˆ’ 1) ∈ ℝ)
72 elfzle2 13529 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑖 βˆ’ 1)) β†’ 𝑗 ≀ (𝑖 βˆ’ 1))
7372adantl 481 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 ≀ (𝑖 βˆ’ 1))
7468ltm1d 12168 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ (𝑖 βˆ’ 1) < 𝑖)
7567, 71, 68, 73, 74lelttrd 11394 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 < 𝑖)
7656adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑖 ≀ 𝑀)
7767, 68, 69, 75, 76ltletrd 11396 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 < 𝑀)
7865adantl 481 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 ∈ β„€)
79 0zd 12592 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 0 ∈ β„€)
8041adantr 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑀 ∈ β„€)
81 elfzo 13658 . . . . . . . . . . 11 ((𝑗 ∈ β„€ ∧ 0 ∈ β„€ ∧ 𝑀 ∈ β„€) β†’ (𝑗 ∈ (0..^𝑀) ↔ (0 ≀ 𝑗 ∧ 𝑗 < 𝑀)))
8278, 79, 80, 81syl3anc 1369 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ (𝑗 ∈ (0..^𝑀) ↔ (0 ≀ 𝑗 ∧ 𝑗 < 𝑀)))
8364, 77, 82mpbir2and 712 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 ∈ (0..^𝑀))
8483adantll 713 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 ∈ (0..^𝑀))
8513adantr 480 . . . . . . . . . 10 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ 𝑄:(0...𝑀)βŸΆβ„)
86 elfzofz 13672 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) β†’ 𝑗 ∈ (0...𝑀))
8786adantl 481 . . . . . . . . . 10 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ 𝑗 ∈ (0...𝑀))
8885, 87ffvelcdmd 7089 . . . . . . . . 9 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ (π‘„β€˜π‘—) ∈ ℝ)
89 fzofzp1 13753 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) β†’ (𝑗 + 1) ∈ (0...𝑀))
9089adantl 481 . . . . . . . . . 10 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ (𝑗 + 1) ∈ (0...𝑀))
9185, 90ffvelcdmd 7089 . . . . . . . . 9 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ (π‘„β€˜(𝑗 + 1)) ∈ ℝ)
92 eleq1w 2811 . . . . . . . . . . . 12 (𝑖 = 𝑗 β†’ (𝑖 ∈ (0..^𝑀) ↔ 𝑗 ∈ (0..^𝑀)))
9392anbi2d 628 . . . . . . . . . . 11 (𝑖 = 𝑗 β†’ ((πœ‘ ∧ 𝑖 ∈ (0..^𝑀)) ↔ (πœ‘ ∧ 𝑗 ∈ (0..^𝑀))))
94 fveq2 6891 . . . . . . . . . . . 12 (𝑖 = 𝑗 β†’ (π‘„β€˜π‘–) = (π‘„β€˜π‘—))
95 oveq1 7421 . . . . . . . . . . . . 13 (𝑖 = 𝑗 β†’ (𝑖 + 1) = (𝑗 + 1))
9695fveq2d 6895 . . . . . . . . . . . 12 (𝑖 = 𝑗 β†’ (π‘„β€˜(𝑖 + 1)) = (π‘„β€˜(𝑗 + 1)))
9794, 96breq12d 5155 . . . . . . . . . . 11 (𝑖 = 𝑗 β†’ ((π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1)) ↔ (π‘„β€˜π‘—) < (π‘„β€˜(𝑗 + 1))))
9893, 97imbi12d 344 . . . . . . . . . 10 (𝑖 = 𝑗 β†’ (((πœ‘ ∧ 𝑖 ∈ (0..^𝑀)) β†’ (π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1))) ↔ ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ (π‘„β€˜π‘—) < (π‘„β€˜(𝑗 + 1)))))
9916simprd 495 . . . . . . . . . . 11 (πœ‘ β†’ βˆ€π‘– ∈ (0..^𝑀)(π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1)))
10099r19.21bi 3243 . . . . . . . . . 10 ((πœ‘ ∧ 𝑖 ∈ (0..^𝑀)) β†’ (π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1)))
10198, 100chvarvv 1995 . . . . . . . . 9 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ (π‘„β€˜π‘—) < (π‘„β€˜(𝑗 + 1)))
10288, 91, 101ltled 11384 . . . . . . . 8 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ (π‘„β€˜π‘—) ≀ (π‘„β€˜(𝑗 + 1)))
10362, 84, 102syl2anc 583 . . . . . . 7 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ (π‘„β€˜π‘—) ≀ (π‘„β€˜(𝑗 + 1)))
10438, 61, 103monoord 14021 . . . . . 6 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ (π‘„β€˜0) ≀ (π‘„β€˜π‘–))
10536, 104eqbrtrd 5164 . . . . 5 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ 𝐴 ≀ (π‘„β€˜π‘–))
106 elfzuz3 13522 . . . . . . . 8 (𝑖 ∈ (0...𝑀) β†’ 𝑀 ∈ (β„€β‰₯β€˜π‘–))
107106adantl 481 . . . . . . 7 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ 𝑀 ∈ (β„€β‰₯β€˜π‘–))
10813ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) β†’ 𝑄:(0...𝑀)βŸΆβ„)
109 fz0fzelfz0 13631 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...𝑀)) β†’ 𝑗 ∈ (0...𝑀))
110109adantll 713 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) β†’ 𝑗 ∈ (0...𝑀))
111108, 110ffvelcdmd 7089 . . . . . . 7 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) β†’ (π‘„β€˜π‘—) ∈ ℝ)
11213ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑄:(0...𝑀)βŸΆβ„)
113 0zd 12592 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 0 ∈ β„€)
11441ad2antlr 726 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑀 ∈ β„€)
115 elfzelz 13525 . . . . . . . . . . 11 (𝑗 ∈ (𝑖...(𝑀 βˆ’ 1)) β†’ 𝑗 ∈ β„€)
116115adantl 481 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ∈ β„€)
117 0red 11239 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 0 ∈ ℝ)
11850adantr 480 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑖 ∈ ℝ)
119115zred 12688 . . . . . . . . . . . . 13 (𝑗 ∈ (𝑖...(𝑀 βˆ’ 1)) β†’ 𝑗 ∈ ℝ)
120119adantl 481 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ∈ ℝ)
121 elfzle1 13528 . . . . . . . . . . . . 13 (𝑖 ∈ (0...𝑀) β†’ 0 ≀ 𝑖)
122121adantr 480 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 0 ≀ 𝑖)
123 elfzle1 13528 . . . . . . . . . . . . 13 (𝑗 ∈ (𝑖...(𝑀 βˆ’ 1)) β†’ 𝑖 ≀ 𝑗)
124123adantl 481 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑖 ≀ 𝑗)
125117, 118, 120, 122, 124letrd 11393 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 0 ≀ 𝑗)
126125adantll 713 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 0 ≀ 𝑗)
127119adantl 481 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ∈ ℝ)
1282nnred 12249 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑀 ∈ ℝ)
129128adantr 480 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑀 ∈ ℝ)
130 1red 11237 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 1 ∈ ℝ)
131129, 130resubcld 11664 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑀 βˆ’ 1) ∈ ℝ)
132 elfzle2 13529 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑖...(𝑀 βˆ’ 1)) β†’ 𝑗 ≀ (𝑀 βˆ’ 1))
133132adantl 481 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ≀ (𝑀 βˆ’ 1))
134129ltm1d 12168 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑀 βˆ’ 1) < 𝑀)
135127, 131, 129, 133, 134lelttrd 11394 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 < 𝑀)
136127, 129, 135ltled 11384 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ≀ 𝑀)
137136adantlr 714 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ≀ 𝑀)
138113, 114, 116, 126, 137elfzd 13516 . . . . . . . . 9 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ∈ (0...𝑀))
139112, 138ffvelcdmd 7089 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (π‘„β€˜π‘—) ∈ ℝ)
140116peano2zd 12691 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑗 + 1) ∈ β„€)
141119adantl 481 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ∈ ℝ)
142 1red 11237 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 1 ∈ ℝ)
143 0le1 11759 . . . . . . . . . . . 12 0 ≀ 1
144143a1i 11 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 0 ≀ 1)
145141, 142, 126, 144addge0d 11812 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 0 ≀ (𝑗 + 1))
146127, 131, 130, 133leadd1dd 11850 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑗 + 1) ≀ ((𝑀 βˆ’ 1) + 1))
1472nncnd 12250 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝑀 ∈ β„‚)
148147adantr 480 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑀 ∈ β„‚)
149 1cnd 11231 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 1 ∈ β„‚)
150148, 149npcand 11597 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ ((𝑀 βˆ’ 1) + 1) = 𝑀)
151146, 150breqtrd 5168 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑗 + 1) ≀ 𝑀)
152151adantlr 714 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑗 + 1) ≀ 𝑀)
153113, 114, 140, 145, 152elfzd 13516 . . . . . . . . 9 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑗 + 1) ∈ (0...𝑀))
154112, 153ffvelcdmd 7089 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (π‘„β€˜(𝑗 + 1)) ∈ ℝ)
155 simpll 766 . . . . . . . . 9 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ πœ‘)
156135adantlr 714 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 < 𝑀)
157116, 113, 114, 81syl3anc 1369 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑗 ∈ (0..^𝑀) ↔ (0 ≀ 𝑗 ∧ 𝑗 < 𝑀)))
158126, 156, 157mpbir2and 712 . . . . . . . . 9 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ∈ (0..^𝑀))
159155, 158, 101syl2anc 583 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (π‘„β€˜π‘—) < (π‘„β€˜(𝑗 + 1)))
160139, 154, 159ltled 11384 . . . . . . 7 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (π‘„β€˜π‘—) ≀ (π‘„β€˜(𝑗 + 1)))
161107, 111, 160monoord 14021 . . . . . 6 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ (π‘„β€˜π‘–) ≀ (π‘„β€˜π‘€))
16228adantr 480 . . . . . 6 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ (π‘„β€˜π‘€) = 𝐡)
163161, 162breqtrd 5168 . . . . 5 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ (π‘„β€˜π‘–) ≀ 𝐡)
16427, 33, 34, 105, 163eliccd 44812 . . . 4 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ (π‘„β€˜π‘–) ∈ (𝐴[,]𝐡))
165164ralrimiva 3141 . . 3 (πœ‘ β†’ βˆ€π‘– ∈ (0...𝑀)(π‘„β€˜π‘–) ∈ (𝐴[,]𝐡))
166 fnfvrnss 7125 . . 3 ((𝑄 Fn (0...𝑀) ∧ βˆ€π‘– ∈ (0...𝑀)(π‘„β€˜π‘–) ∈ (𝐴[,]𝐡)) β†’ ran 𝑄 βŠ† (𝐴[,]𝐡))
16715, 165, 166syl2anc 583 . 2 (πœ‘ β†’ ran 𝑄 βŠ† (𝐴[,]𝐡))
168 df-f 6546 . 2 (𝑄:(0...𝑀)⟢(𝐴[,]𝐡) ↔ (𝑄 Fn (0...𝑀) ∧ ran 𝑄 βŠ† (𝐴[,]𝐡)))
16915, 167, 168sylanbrc 582 1 (πœ‘ β†’ 𝑄:(0...𝑀)⟢(𝐴[,]𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3056  {crab 3427  Vcvv 3469   βŠ† wss 3944   class class class wbr 5142   ↦ cmpt 5225  ran crn 5673   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7414   ↑m cmap 8836  β„‚cc 11128  β„cr 11129  0cc0 11130  1c1 11131   + caddc 11133   < clt 11270   ≀ cle 11271   βˆ’ cmin 11466  β„•cn 12234  β„•0cn0 12494  β„€cz 12580  β„€β‰₯cuz 12844  [,]cicc 13351  ...cfz 13508  ..^cfzo 13651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-map 8838  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-n0 12495  df-z 12581  df-uz 12845  df-icc 13355  df-fz 13509  df-fzo 13652
This theorem is referenced by:  fourierdlem38  45456  fourierdlem50  45467  fourierdlem54  45471  fourierdlem63  45480  fourierdlem65  45482  fourierdlem69  45486  fourierdlem70  45487  fourierdlem74  45491  fourierdlem75  45492  fourierdlem76  45493  fourierdlem79  45496  fourierdlem81  45498  fourierdlem84  45501  fourierdlem85  45502  fourierdlem88  45505  fourierdlem89  45506  fourierdlem90  45507  fourierdlem91  45508  fourierdlem92  45509  fourierdlem93  45510  fourierdlem100  45517  fourierdlem101  45518  fourierdlem103  45520  fourierdlem104  45521  fourierdlem107  45524  fourierdlem111  45528  fourierdlem112  45529  fourierdlem113  45530
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