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Theorem fourierdlem15 45569
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 45556 . . . . . . 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 493 . . . 4 (πœ‘ β†’ 𝑄 ∈ (ℝ ↑m (0...𝑀)))
8 reex 11224 . . . . . 6 ℝ ∈ V
98a1i 11 . . . . 5 (πœ‘ β†’ ℝ ∈ V)
10 ovex 7446 . . . . . 6 (0...𝑀) ∈ V
1110a1i 11 . . . . 5 (πœ‘ β†’ (0...𝑀) ∈ V)
129, 11elmapd 8852 . . . 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 6717 . . 3 (𝑄:(0...𝑀)βŸΆβ„ β†’ 𝑄 Fn (0...𝑀))
1513, 14syl 17 . 2 (πœ‘ β†’ 𝑄 Fn (0...𝑀))
166simprd 494 . . . . . . . . 9 (πœ‘ β†’ (((π‘„β€˜0) = 𝐴 ∧ (π‘„β€˜π‘€) = 𝐡) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1))))
1716simpld 493 . . . . . . . 8 (πœ‘ β†’ ((π‘„β€˜0) = 𝐴 ∧ (π‘„β€˜π‘€) = 𝐡))
1817simpld 493 . . . . . . 7 (πœ‘ β†’ (π‘„β€˜0) = 𝐴)
19 nnnn0 12504 . . . . . . . . . . 11 (𝑀 ∈ β„• β†’ 𝑀 ∈ β„•0)
20 nn0uz 12889 . . . . . . . . . . 11 β„•0 = (β„€β‰₯β€˜0)
2119, 20eleqtrdi 2835 . . . . . . . . . 10 (𝑀 ∈ β„• β†’ 𝑀 ∈ (β„€β‰₯β€˜0))
222, 21syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝑀 ∈ (β„€β‰₯β€˜0))
23 eluzfz1 13535 . . . . . . . . 9 (𝑀 ∈ (β„€β‰₯β€˜0) β†’ 0 ∈ (0...𝑀))
2422, 23syl 17 . . . . . . . 8 (πœ‘ β†’ 0 ∈ (0...𝑀))
2513, 24ffvelcdmd 7088 . . . . . . 7 (πœ‘ β†’ (π‘„β€˜0) ∈ ℝ)
2618, 25eqeltrrd 2826 . . . . . 6 (πœ‘ β†’ 𝐴 ∈ ℝ)
2726adantr 479 . . . . 5 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ 𝐴 ∈ ℝ)
2817simprd 494 . . . . . . 7 (πœ‘ β†’ (π‘„β€˜π‘€) = 𝐡)
29 eluzfz2 13536 . . . . . . . . 9 (𝑀 ∈ (β„€β‰₯β€˜0) β†’ 𝑀 ∈ (0...𝑀))
3022, 29syl 17 . . . . . . . 8 (πœ‘ β†’ 𝑀 ∈ (0...𝑀))
3113, 30ffvelcdmd 7088 . . . . . . 7 (πœ‘ β†’ (π‘„β€˜π‘€) ∈ ℝ)
3228, 31eqeltrrd 2826 . . . . . 6 (πœ‘ β†’ 𝐡 ∈ ℝ)
3332adantr 479 . . . . 5 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ 𝐡 ∈ ℝ)
3413ffvelcdmda 7087 . . . . 5 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ (π‘„β€˜π‘–) ∈ ℝ)
3518eqcomd 2731 . . . . . . 7 (πœ‘ β†’ 𝐴 = (π‘„β€˜0))
3635adantr 479 . . . . . 6 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ 𝐴 = (π‘„β€˜0))
37 elfzuz 13524 . . . . . . . 8 (𝑖 ∈ (0...𝑀) β†’ 𝑖 ∈ (β„€β‰₯β€˜0))
3837adantl 480 . . . . . . 7 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ 𝑖 ∈ (β„€β‰₯β€˜0))
3913ad2antrr 724 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑄:(0...𝑀)βŸΆβ„)
40 0zd 12595 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 0 ∈ β„€)
41 elfzel2 13526 . . . . . . . . . . 11 (𝑖 ∈ (0...𝑀) β†’ 𝑀 ∈ β„€)
4241adantr 479 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑀 ∈ β„€)
43 elfzelz 13528 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑖) β†’ 𝑗 ∈ β„€)
4443adantl 480 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑗 ∈ β„€)
45 elfzle1 13531 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑖) β†’ 0 ≀ 𝑗)
4645adantl 480 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 0 ≀ 𝑗)
4743zred 12691 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑖) β†’ 𝑗 ∈ ℝ)
4847adantl 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑗 ∈ ℝ)
49 elfzelz 13528 . . . . . . . . . . . . 13 (𝑖 ∈ (0...𝑀) β†’ 𝑖 ∈ β„€)
5049zred 12691 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) β†’ 𝑖 ∈ ℝ)
5150adantr 479 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑖 ∈ ℝ)
5241zred 12691 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) β†’ 𝑀 ∈ ℝ)
5352adantr 479 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑀 ∈ ℝ)
54 elfzle2 13532 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑖) β†’ 𝑗 ≀ 𝑖)
5554adantl 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑗 ≀ 𝑖)
56 elfzle2 13532 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) β†’ 𝑖 ≀ 𝑀)
5756adantr 479 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑖 ≀ 𝑀)
5848, 51, 53, 55, 57letrd 11396 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑗 ≀ 𝑀)
5940, 42, 44, 46, 58elfzd 13519 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑗 ∈ (0...𝑀))
6059adantll 712 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑗 ∈ (0...𝑀))
6139, 60ffvelcdmd 7088 . . . . . . 7 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...𝑖)) β†’ (π‘„β€˜π‘—) ∈ ℝ)
62 simpll 765 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ πœ‘)
63 elfzle1 13531 . . . . . . . . . . 11 (𝑗 ∈ (0...(𝑖 βˆ’ 1)) β†’ 0 ≀ 𝑗)
6463adantl 480 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 0 ≀ 𝑗)
65 elfzelz 13528 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑖 βˆ’ 1)) β†’ 𝑗 ∈ β„€)
6665zred 12691 . . . . . . . . . . . 12 (𝑗 ∈ (0...(𝑖 βˆ’ 1)) β†’ 𝑗 ∈ ℝ)
6766adantl 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 ∈ ℝ)
6850adantr 479 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑖 ∈ ℝ)
6952adantr 479 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑀 ∈ ℝ)
70 peano2rem 11552 . . . . . . . . . . . . 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 13532 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑖 βˆ’ 1)) β†’ 𝑗 ≀ (𝑖 βˆ’ 1))
7372adantl 480 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 ≀ (𝑖 βˆ’ 1))
7468ltm1d 12171 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ (𝑖 βˆ’ 1) < 𝑖)
7567, 71, 68, 73, 74lelttrd 11397 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 < 𝑖)
7656adantr 479 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑖 ≀ 𝑀)
7767, 68, 69, 75, 76ltletrd 11399 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 < 𝑀)
7865adantl 480 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 ∈ β„€)
79 0zd 12595 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 0 ∈ β„€)
8041adantr 479 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑀 ∈ β„€)
81 elfzo 13661 . . . . . . . . . . 11 ((𝑗 ∈ β„€ ∧ 0 ∈ β„€ ∧ 𝑀 ∈ β„€) β†’ (𝑗 ∈ (0..^𝑀) ↔ (0 ≀ 𝑗 ∧ 𝑗 < 𝑀)))
8278, 79, 80, 81syl3anc 1368 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ (𝑗 ∈ (0..^𝑀) ↔ (0 ≀ 𝑗 ∧ 𝑗 < 𝑀)))
8364, 77, 82mpbir2and 711 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 ∈ (0..^𝑀))
8483adantll 712 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 ∈ (0..^𝑀))
8513adantr 479 . . . . . . . . . 10 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ 𝑄:(0...𝑀)βŸΆβ„)
86 elfzofz 13675 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) β†’ 𝑗 ∈ (0...𝑀))
8786adantl 480 . . . . . . . . . 10 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ 𝑗 ∈ (0...𝑀))
8885, 87ffvelcdmd 7088 . . . . . . . . 9 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ (π‘„β€˜π‘—) ∈ ℝ)
89 fzofzp1 13756 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) β†’ (𝑗 + 1) ∈ (0...𝑀))
9089adantl 480 . . . . . . . . . 10 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ (𝑗 + 1) ∈ (0...𝑀))
9185, 90ffvelcdmd 7088 . . . . . . . . 9 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ (π‘„β€˜(𝑗 + 1)) ∈ ℝ)
92 eleq1w 2808 . . . . . . . . . . . 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 6890 . . . . . . . . . . . 12 (𝑖 = 𝑗 β†’ (π‘„β€˜π‘–) = (π‘„β€˜π‘—))
95 oveq1 7420 . . . . . . . . . . . . 13 (𝑖 = 𝑗 β†’ (𝑖 + 1) = (𝑗 + 1))
9695fveq2d 6894 . . . . . . . . . . . 12 (𝑖 = 𝑗 β†’ (π‘„β€˜(𝑖 + 1)) = (π‘„β€˜(𝑗 + 1)))
9794, 96breq12d 5157 . . . . . . . . . . 11 (𝑖 = 𝑗 β†’ ((π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1)) ↔ (π‘„β€˜π‘—) < (π‘„β€˜(𝑗 + 1))))
9893, 97imbi12d 343 . . . . . . . . . 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 494 . . . . . . . . . . 11 (πœ‘ β†’ βˆ€π‘– ∈ (0..^𝑀)(π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1)))
10099r19.21bi 3239 . . . . . . . . . 10 ((πœ‘ ∧ 𝑖 ∈ (0..^𝑀)) β†’ (π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1)))
10198, 100chvarvv 1994 . . . . . . . . 9 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ (π‘„β€˜π‘—) < (π‘„β€˜(𝑗 + 1)))
10288, 91, 101ltled 11387 . . . . . . . 8 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ (π‘„β€˜π‘—) ≀ (π‘„β€˜(𝑗 + 1)))
10362, 84, 102syl2anc 582 . . . . . . 7 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ (π‘„β€˜π‘—) ≀ (π‘„β€˜(𝑗 + 1)))
10438, 61, 103monoord 14024 . . . . . 6 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ (π‘„β€˜0) ≀ (π‘„β€˜π‘–))
10536, 104eqbrtrd 5166 . . . . 5 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ 𝐴 ≀ (π‘„β€˜π‘–))
106 elfzuz3 13525 . . . . . . . 8 (𝑖 ∈ (0...𝑀) β†’ 𝑀 ∈ (β„€β‰₯β€˜π‘–))
107106adantl 480 . . . . . . 7 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ 𝑀 ∈ (β„€β‰₯β€˜π‘–))
10813ad2antrr 724 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) β†’ 𝑄:(0...𝑀)βŸΆβ„)
109 fz0fzelfz0 13634 . . . . . . . . 9 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...𝑀)) β†’ 𝑗 ∈ (0...𝑀))
110109adantll 712 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) β†’ 𝑗 ∈ (0...𝑀))
111108, 110ffvelcdmd 7088 . . . . . . 7 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...𝑀)) β†’ (π‘„β€˜π‘—) ∈ ℝ)
11213ad2antrr 724 . . . . . . . . 9 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑄:(0...𝑀)βŸΆβ„)
113 0zd 12595 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 0 ∈ β„€)
11441ad2antlr 725 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑀 ∈ β„€)
115 elfzelz 13528 . . . . . . . . . . 11 (𝑗 ∈ (𝑖...(𝑀 βˆ’ 1)) β†’ 𝑗 ∈ β„€)
116115adantl 480 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ∈ β„€)
117 0red 11242 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 0 ∈ ℝ)
11850adantr 479 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑖 ∈ ℝ)
119115zred 12691 . . . . . . . . . . . . 13 (𝑗 ∈ (𝑖...(𝑀 βˆ’ 1)) β†’ 𝑗 ∈ ℝ)
120119adantl 480 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ∈ ℝ)
121 elfzle1 13531 . . . . . . . . . . . . 13 (𝑖 ∈ (0...𝑀) β†’ 0 ≀ 𝑖)
122121adantr 479 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 0 ≀ 𝑖)
123 elfzle1 13531 . . . . . . . . . . . . 13 (𝑗 ∈ (𝑖...(𝑀 βˆ’ 1)) β†’ 𝑖 ≀ 𝑗)
124123adantl 480 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑖 ≀ 𝑗)
125117, 118, 120, 122, 124letrd 11396 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 0 ≀ 𝑗)
126125adantll 712 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 0 ≀ 𝑗)
127119adantl 480 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ∈ ℝ)
1282nnred 12252 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑀 ∈ ℝ)
129128adantr 479 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑀 ∈ ℝ)
130 1red 11240 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 1 ∈ ℝ)
131129, 130resubcld 11667 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑀 βˆ’ 1) ∈ ℝ)
132 elfzle2 13532 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑖...(𝑀 βˆ’ 1)) β†’ 𝑗 ≀ (𝑀 βˆ’ 1))
133132adantl 480 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ≀ (𝑀 βˆ’ 1))
134129ltm1d 12171 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑀 βˆ’ 1) < 𝑀)
135127, 131, 129, 133, 134lelttrd 11397 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 < 𝑀)
136127, 129, 135ltled 11387 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ≀ 𝑀)
137136adantlr 713 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ≀ 𝑀)
138113, 114, 116, 126, 137elfzd 13519 . . . . . . . . 9 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ∈ (0...𝑀))
139112, 138ffvelcdmd 7088 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (π‘„β€˜π‘—) ∈ ℝ)
140116peano2zd 12694 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑗 + 1) ∈ β„€)
141119adantl 480 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ∈ ℝ)
142 1red 11240 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 1 ∈ ℝ)
143 0le1 11762 . . . . . . . . . . . 12 0 ≀ 1
144143a1i 11 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 0 ≀ 1)
145141, 142, 126, 144addge0d 11815 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 0 ≀ (𝑗 + 1))
146127, 131, 130, 133leadd1dd 11853 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑗 + 1) ≀ ((𝑀 βˆ’ 1) + 1))
1472nncnd 12253 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝑀 ∈ β„‚)
148147adantr 479 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑀 ∈ β„‚)
149 1cnd 11234 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 1 ∈ β„‚)
150148, 149npcand 11600 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ ((𝑀 βˆ’ 1) + 1) = 𝑀)
151146, 150breqtrd 5170 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑗 + 1) ≀ 𝑀)
152151adantlr 713 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑗 + 1) ≀ 𝑀)
153113, 114, 140, 145, 152elfzd 13519 . . . . . . . . 9 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑗 + 1) ∈ (0...𝑀))
154112, 153ffvelcdmd 7088 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (π‘„β€˜(𝑗 + 1)) ∈ ℝ)
155 simpll 765 . . . . . . . . 9 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ πœ‘)
156135adantlr 713 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 < 𝑀)
157116, 113, 114, 81syl3anc 1368 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑗 ∈ (0..^𝑀) ↔ (0 ≀ 𝑗 ∧ 𝑗 < 𝑀)))
158126, 156, 157mpbir2and 711 . . . . . . . . 9 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ∈ (0..^𝑀))
159155, 158, 101syl2anc 582 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (π‘„β€˜π‘—) < (π‘„β€˜(𝑗 + 1)))
160139, 154, 159ltled 11387 . . . . . . 7 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (π‘„β€˜π‘—) ≀ (π‘„β€˜(𝑗 + 1)))
161107, 111, 160monoord 14024 . . . . . 6 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ (π‘„β€˜π‘–) ≀ (π‘„β€˜π‘€))
16228adantr 479 . . . . . 6 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ (π‘„β€˜π‘€) = 𝐡)
163161, 162breqtrd 5170 . . . . 5 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ (π‘„β€˜π‘–) ≀ 𝐡)
16427, 33, 34, 105, 163eliccd 44948 . . . 4 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ (π‘„β€˜π‘–) ∈ (𝐴[,]𝐡))
165164ralrimiva 3136 . . 3 (πœ‘ β†’ βˆ€π‘– ∈ (0...𝑀)(π‘„β€˜π‘–) ∈ (𝐴[,]𝐡))
166 fnfvrnss 7124 . . 3 ((𝑄 Fn (0...𝑀) ∧ βˆ€π‘– ∈ (0...𝑀)(π‘„β€˜π‘–) ∈ (𝐴[,]𝐡)) β†’ ran 𝑄 βŠ† (𝐴[,]𝐡))
16715, 165, 166syl2anc 582 . 2 (πœ‘ β†’ ran 𝑄 βŠ† (𝐴[,]𝐡))
168 df-f 6547 . 2 (𝑄:(0...𝑀)⟢(𝐴[,]𝐡) ↔ (𝑄 Fn (0...𝑀) ∧ ran 𝑄 βŠ† (𝐴[,]𝐡)))
16915, 167, 168sylanbrc 581 1 (πœ‘ β†’ 𝑄:(0...𝑀)⟢(𝐴[,]𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  {crab 3419  Vcvv 3463   βŠ† wss 3941   class class class wbr 5144   ↦ cmpt 5227  ran crn 5674   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7413   ↑m cmap 8838  β„‚cc 11131  β„cr 11132  0cc0 11133  1c1 11134   + caddc 11136   < clt 11273   ≀ cle 11274   βˆ’ cmin 11469  β„•cn 12237  β„•0cn0 12497  β„€cz 12583  β„€β‰₯cuz 12847  [,]cicc 13354  ...cfz 13511  ..^cfzo 13654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-nn 12238  df-n0 12498  df-z 12584  df-uz 12848  df-icc 13358  df-fz 13512  df-fzo 13655
This theorem is referenced by:  fourierdlem38  45592  fourierdlem50  45603  fourierdlem54  45607  fourierdlem63  45616  fourierdlem65  45618  fourierdlem69  45622  fourierdlem70  45623  fourierdlem74  45627  fourierdlem75  45628  fourierdlem76  45629  fourierdlem79  45632  fourierdlem81  45634  fourierdlem84  45637  fourierdlem85  45638  fourierdlem88  45641  fourierdlem89  45642  fourierdlem90  45643  fourierdlem91  45644  fourierdlem92  45645  fourierdlem93  45646  fourierdlem100  45653  fourierdlem101  45654  fourierdlem103  45656  fourierdlem104  45657  fourierdlem107  45660  fourierdlem111  45664  fourierdlem112  45665  fourierdlem113  45666
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