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Theorem fourierdlem15 44449
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 44436 . . . . . . 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 496 . . . 4 (πœ‘ β†’ 𝑄 ∈ (ℝ ↑m (0...𝑀)))
8 reex 11147 . . . . . 6 ℝ ∈ V
98a1i 11 . . . . 5 (πœ‘ β†’ ℝ ∈ V)
10 ovex 7391 . . . . . 6 (0...𝑀) ∈ V
1110a1i 11 . . . . 5 (πœ‘ β†’ (0...𝑀) ∈ V)
129, 11elmapd 8782 . . . 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 6669 . . 3 (𝑄:(0...𝑀)βŸΆβ„ β†’ 𝑄 Fn (0...𝑀))
1513, 14syl 17 . 2 (πœ‘ β†’ 𝑄 Fn (0...𝑀))
166simprd 497 . . . . . . . . 9 (πœ‘ β†’ (((π‘„β€˜0) = 𝐴 ∧ (π‘„β€˜π‘€) = 𝐡) ∧ βˆ€π‘– ∈ (0..^𝑀)(π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1))))
1716simpld 496 . . . . . . . 8 (πœ‘ β†’ ((π‘„β€˜0) = 𝐴 ∧ (π‘„β€˜π‘€) = 𝐡))
1817simpld 496 . . . . . . 7 (πœ‘ β†’ (π‘„β€˜0) = 𝐴)
19 nnnn0 12425 . . . . . . . . . . 11 (𝑀 ∈ β„• β†’ 𝑀 ∈ β„•0)
20 nn0uz 12810 . . . . . . . . . . 11 β„•0 = (β„€β‰₯β€˜0)
2119, 20eleqtrdi 2844 . . . . . . . . . 10 (𝑀 ∈ β„• β†’ 𝑀 ∈ (β„€β‰₯β€˜0))
222, 21syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝑀 ∈ (β„€β‰₯β€˜0))
23 eluzfz1 13454 . . . . . . . . 9 (𝑀 ∈ (β„€β‰₯β€˜0) β†’ 0 ∈ (0...𝑀))
2422, 23syl 17 . . . . . . . 8 (πœ‘ β†’ 0 ∈ (0...𝑀))
2513, 24ffvelcdmd 7037 . . . . . . 7 (πœ‘ β†’ (π‘„β€˜0) ∈ ℝ)
2618, 25eqeltrrd 2835 . . . . . 6 (πœ‘ β†’ 𝐴 ∈ ℝ)
2726adantr 482 . . . . 5 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ 𝐴 ∈ ℝ)
2817simprd 497 . . . . . . 7 (πœ‘ β†’ (π‘„β€˜π‘€) = 𝐡)
29 eluzfz2 13455 . . . . . . . . 9 (𝑀 ∈ (β„€β‰₯β€˜0) β†’ 𝑀 ∈ (0...𝑀))
3022, 29syl 17 . . . . . . . 8 (πœ‘ β†’ 𝑀 ∈ (0...𝑀))
3113, 30ffvelcdmd 7037 . . . . . . 7 (πœ‘ β†’ (π‘„β€˜π‘€) ∈ ℝ)
3228, 31eqeltrrd 2835 . . . . . 6 (πœ‘ β†’ 𝐡 ∈ ℝ)
3332adantr 482 . . . . 5 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ 𝐡 ∈ ℝ)
3413ffvelcdmda 7036 . . . . 5 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ (π‘„β€˜π‘–) ∈ ℝ)
3518eqcomd 2739 . . . . . . 7 (πœ‘ β†’ 𝐴 = (π‘„β€˜0))
3635adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ 𝐴 = (π‘„β€˜0))
37 elfzuz 13443 . . . . . . . 8 (𝑖 ∈ (0...𝑀) β†’ 𝑖 ∈ (β„€β‰₯β€˜0))
3837adantl 483 . . . . . . 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 12516 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 0 ∈ β„€)
41 elfzel2 13445 . . . . . . . . . . 11 (𝑖 ∈ (0...𝑀) β†’ 𝑀 ∈ β„€)
4241adantr 482 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑀 ∈ β„€)
43 elfzelz 13447 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑖) β†’ 𝑗 ∈ β„€)
4443adantl 483 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑗 ∈ β„€)
45 elfzle1 13450 . . . . . . . . . . 11 (𝑗 ∈ (0...𝑖) β†’ 0 ≀ 𝑗)
4645adantl 483 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 0 ≀ 𝑗)
4743zred 12612 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑖) β†’ 𝑗 ∈ ℝ)
4847adantl 483 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑗 ∈ ℝ)
49 elfzelz 13447 . . . . . . . . . . . . 13 (𝑖 ∈ (0...𝑀) β†’ 𝑖 ∈ β„€)
5049zred 12612 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) β†’ 𝑖 ∈ ℝ)
5150adantr 482 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑖 ∈ ℝ)
5241zred 12612 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) β†’ 𝑀 ∈ ℝ)
5352adantr 482 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑀 ∈ ℝ)
54 elfzle2 13451 . . . . . . . . . . . 12 (𝑗 ∈ (0...𝑖) β†’ 𝑗 ≀ 𝑖)
5554adantl 483 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑗 ≀ 𝑖)
56 elfzle2 13451 . . . . . . . . . . . 12 (𝑖 ∈ (0...𝑀) β†’ 𝑖 ≀ 𝑀)
5756adantr 482 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑖 ≀ 𝑀)
5848, 51, 53, 55, 57letrd 11317 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑖)) β†’ 𝑗 ≀ 𝑀)
5940, 42, 44, 46, 58elfzd 13438 . . . . . . . . 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 7037 . . . . . . 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 13450 . . . . . . . . . . 11 (𝑗 ∈ (0...(𝑖 βˆ’ 1)) β†’ 0 ≀ 𝑗)
6463adantl 483 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 0 ≀ 𝑗)
65 elfzelz 13447 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑖 βˆ’ 1)) β†’ 𝑗 ∈ β„€)
6665zred 12612 . . . . . . . . . . . 12 (𝑗 ∈ (0...(𝑖 βˆ’ 1)) β†’ 𝑗 ∈ ℝ)
6766adantl 483 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 ∈ ℝ)
6850adantr 482 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑖 ∈ ℝ)
6952adantr 482 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑀 ∈ ℝ)
70 peano2rem 11473 . . . . . . . . . . . . 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 13451 . . . . . . . . . . . . 13 (𝑗 ∈ (0...(𝑖 βˆ’ 1)) β†’ 𝑗 ≀ (𝑖 βˆ’ 1))
7372adantl 483 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 ≀ (𝑖 βˆ’ 1))
7468ltm1d 12092 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ (𝑖 βˆ’ 1) < 𝑖)
7567, 71, 68, 73, 74lelttrd 11318 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 < 𝑖)
7656adantr 482 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑖 ≀ 𝑀)
7767, 68, 69, 75, 76ltletrd 11320 . . . . . . . . . 10 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 < 𝑀)
7865adantl 483 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑗 ∈ β„€)
79 0zd 12516 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 0 ∈ β„€)
8041adantr 482 . . . . . . . . . . 11 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ 𝑀 ∈ β„€)
81 elfzo 13580 . . . . . . . . . . 11 ((𝑗 ∈ β„€ ∧ 0 ∈ β„€ ∧ 𝑀 ∈ β„€) β†’ (𝑗 ∈ (0..^𝑀) ↔ (0 ≀ 𝑗 ∧ 𝑗 < 𝑀)))
8278, 79, 80, 81syl3anc 1372 . . . . . . . . . 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 482 . . . . . . . . . 10 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ 𝑄:(0...𝑀)βŸΆβ„)
86 elfzofz 13594 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) β†’ 𝑗 ∈ (0...𝑀))
8786adantl 483 . . . . . . . . . 10 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ 𝑗 ∈ (0...𝑀))
8885, 87ffvelcdmd 7037 . . . . . . . . 9 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ (π‘„β€˜π‘—) ∈ ℝ)
89 fzofzp1 13675 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) β†’ (𝑗 + 1) ∈ (0...𝑀))
9089adantl 483 . . . . . . . . . 10 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ (𝑗 + 1) ∈ (0...𝑀))
9185, 90ffvelcdmd 7037 . . . . . . . . 9 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ (π‘„β€˜(𝑗 + 1)) ∈ ℝ)
92 eleq1w 2817 . . . . . . . . . . . 12 (𝑖 = 𝑗 β†’ (𝑖 ∈ (0..^𝑀) ↔ 𝑗 ∈ (0..^𝑀)))
9392anbi2d 630 . . . . . . . . . . 11 (𝑖 = 𝑗 β†’ ((πœ‘ ∧ 𝑖 ∈ (0..^𝑀)) ↔ (πœ‘ ∧ 𝑗 ∈ (0..^𝑀))))
94 fveq2 6843 . . . . . . . . . . . 12 (𝑖 = 𝑗 β†’ (π‘„β€˜π‘–) = (π‘„β€˜π‘—))
95 oveq1 7365 . . . . . . . . . . . . 13 (𝑖 = 𝑗 β†’ (𝑖 + 1) = (𝑗 + 1))
9695fveq2d 6847 . . . . . . . . . . . 12 (𝑖 = 𝑗 β†’ (π‘„β€˜(𝑖 + 1)) = (π‘„β€˜(𝑗 + 1)))
9794, 96breq12d 5119 . . . . . . . . . . 11 (𝑖 = 𝑗 β†’ ((π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1)) ↔ (π‘„β€˜π‘—) < (π‘„β€˜(𝑗 + 1))))
9893, 97imbi12d 345 . . . . . . . . . 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 497 . . . . . . . . . . 11 (πœ‘ β†’ βˆ€π‘– ∈ (0..^𝑀)(π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1)))
10099r19.21bi 3233 . . . . . . . . . 10 ((πœ‘ ∧ 𝑖 ∈ (0..^𝑀)) β†’ (π‘„β€˜π‘–) < (π‘„β€˜(𝑖 + 1)))
10198, 100chvarvv 2003 . . . . . . . . 9 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ (π‘„β€˜π‘—) < (π‘„β€˜(𝑗 + 1)))
10288, 91, 101ltled 11308 . . . . . . . 8 ((πœ‘ ∧ 𝑗 ∈ (0..^𝑀)) β†’ (π‘„β€˜π‘—) ≀ (π‘„β€˜(𝑗 + 1)))
10362, 84, 102syl2anc 585 . . . . . . 7 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (0...(𝑖 βˆ’ 1))) β†’ (π‘„β€˜π‘—) ≀ (π‘„β€˜(𝑗 + 1)))
10438, 61, 103monoord 13944 . . . . . 6 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ (π‘„β€˜0) ≀ (π‘„β€˜π‘–))
10536, 104eqbrtrd 5128 . . . . 5 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ 𝐴 ≀ (π‘„β€˜π‘–))
106 elfzuz3 13444 . . . . . . . 8 (𝑖 ∈ (0...𝑀) β†’ 𝑀 ∈ (β„€β‰₯β€˜π‘–))
107106adantl 483 . . . . . . 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 13553 . . . . . . . . 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 7037 . . . . . . 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 12516 . . . . . . . . . 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 13447 . . . . . . . . . . 11 (𝑗 ∈ (𝑖...(𝑀 βˆ’ 1)) β†’ 𝑗 ∈ β„€)
116115adantl 483 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ∈ β„€)
117 0red 11163 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 0 ∈ ℝ)
11850adantr 482 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑖 ∈ ℝ)
119115zred 12612 . . . . . . . . . . . . 13 (𝑗 ∈ (𝑖...(𝑀 βˆ’ 1)) β†’ 𝑗 ∈ ℝ)
120119adantl 483 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ∈ ℝ)
121 elfzle1 13450 . . . . . . . . . . . . 13 (𝑖 ∈ (0...𝑀) β†’ 0 ≀ 𝑖)
122121adantr 482 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 0 ≀ 𝑖)
123 elfzle1 13450 . . . . . . . . . . . . 13 (𝑗 ∈ (𝑖...(𝑀 βˆ’ 1)) β†’ 𝑖 ≀ 𝑗)
124123adantl 483 . . . . . . . . . . . 12 ((𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑖 ≀ 𝑗)
125117, 118, 120, 122, 124letrd 11317 . . . . . . . . . . 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 483 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ∈ ℝ)
1282nnred 12173 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝑀 ∈ ℝ)
129128adantr 482 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑀 ∈ ℝ)
130 1red 11161 . . . . . . . . . . . . . 14 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 1 ∈ ℝ)
131129, 130resubcld 11588 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑀 βˆ’ 1) ∈ ℝ)
132 elfzle2 13451 . . . . . . . . . . . . . 14 (𝑗 ∈ (𝑖...(𝑀 βˆ’ 1)) β†’ 𝑗 ≀ (𝑀 βˆ’ 1))
133132adantl 483 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ≀ (𝑀 βˆ’ 1))
134129ltm1d 12092 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑀 βˆ’ 1) < 𝑀)
135127, 131, 129, 133, 134lelttrd 11318 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 < 𝑀)
136127, 129, 135ltled 11308 . . . . . . . . . . 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 13438 . . . . . . . . 9 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ∈ (0...𝑀))
139112, 138ffvelcdmd 7037 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (π‘„β€˜π‘—) ∈ ℝ)
140116peano2zd 12615 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑗 + 1) ∈ β„€)
141119adantl 483 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑗 ∈ ℝ)
142 1red 11161 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 1 ∈ ℝ)
143 0le1 11683 . . . . . . . . . . . 12 0 ≀ 1
144143a1i 11 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 0 ≀ 1)
145141, 142, 126, 144addge0d 11736 . . . . . . . . . 10 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 0 ≀ (𝑗 + 1))
146127, 131, 130, 133leadd1dd 11774 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑗 + 1) ≀ ((𝑀 βˆ’ 1) + 1))
1472nncnd 12174 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝑀 ∈ β„‚)
148147adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 𝑀 ∈ β„‚)
149 1cnd 11155 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ 1 ∈ β„‚)
150148, 149npcand 11521 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ ((𝑀 βˆ’ 1) + 1) = 𝑀)
151146, 150breqtrd 5132 . . . . . . . . . . 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 13438 . . . . . . . . 9 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (𝑗 + 1) ∈ (0...𝑀))
154112, 153ffvelcdmd 7037 . . . . . . . 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 1372 . . . . . . . . . 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 585 . . . . . . . 8 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (π‘„β€˜π‘—) < (π‘„β€˜(𝑗 + 1)))
160139, 154, 159ltled 11308 . . . . . . 7 (((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) ∧ 𝑗 ∈ (𝑖...(𝑀 βˆ’ 1))) β†’ (π‘„β€˜π‘—) ≀ (π‘„β€˜(𝑗 + 1)))
161107, 111, 160monoord 13944 . . . . . 6 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ (π‘„β€˜π‘–) ≀ (π‘„β€˜π‘€))
16228adantr 482 . . . . . 6 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ (π‘„β€˜π‘€) = 𝐡)
163161, 162breqtrd 5132 . . . . 5 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ (π‘„β€˜π‘–) ≀ 𝐡)
16427, 33, 34, 105, 163eliccd 43828 . . . 4 ((πœ‘ ∧ 𝑖 ∈ (0...𝑀)) β†’ (π‘„β€˜π‘–) ∈ (𝐴[,]𝐡))
165164ralrimiva 3140 . . 3 (πœ‘ β†’ βˆ€π‘– ∈ (0...𝑀)(π‘„β€˜π‘–) ∈ (𝐴[,]𝐡))
166 fnfvrnss 7069 . . 3 ((𝑄 Fn (0...𝑀) ∧ βˆ€π‘– ∈ (0...𝑀)(π‘„β€˜π‘–) ∈ (𝐴[,]𝐡)) β†’ ran 𝑄 βŠ† (𝐴[,]𝐡))
16715, 165, 166syl2anc 585 . 2 (πœ‘ β†’ ran 𝑄 βŠ† (𝐴[,]𝐡))
168 df-f 6501 . 2 (𝑄:(0...𝑀)⟢(𝐴[,]𝐡) ↔ (𝑄 Fn (0...𝑀) ∧ ran 𝑄 βŠ† (𝐴[,]𝐡)))
16915, 167, 168sylanbrc 584 1 (πœ‘ β†’ 𝑄:(0...𝑀)⟢(𝐴[,]𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  {crab 3406  Vcvv 3444   βŠ† wss 3911   class class class wbr 5106   ↦ cmpt 5189  ran crn 5635   Fn wfn 6492  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358   ↑m cmap 8768  β„‚cc 11054  β„cr 11055  0cc0 11056  1c1 11057   + caddc 11059   < clt 11194   ≀ cle 11195   βˆ’ cmin 11390  β„•cn 12158  β„•0cn0 12418  β„€cz 12504  β„€β‰₯cuz 12768  [,]cicc 13273  ...cfz 13430  ..^cfzo 13573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-n0 12419  df-z 12505  df-uz 12769  df-icc 13277  df-fz 13431  df-fzo 13574
This theorem is referenced by:  fourierdlem38  44472  fourierdlem50  44483  fourierdlem54  44487  fourierdlem63  44496  fourierdlem65  44498  fourierdlem69  44502  fourierdlem70  44503  fourierdlem74  44507  fourierdlem75  44508  fourierdlem76  44509  fourierdlem79  44512  fourierdlem81  44514  fourierdlem84  44517  fourierdlem85  44518  fourierdlem88  44521  fourierdlem89  44522  fourierdlem90  44523  fourierdlem91  44524  fourierdlem92  44525  fourierdlem93  44526  fourierdlem100  44533  fourierdlem101  44534  fourierdlem103  44536  fourierdlem104  44537  fourierdlem107  44540  fourierdlem111  44544  fourierdlem112  44545  fourierdlem113  44546
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