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Theorem fourierdlem17 39017
Description: The defined 𝐿 is actually a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
fourierdlem17.a (𝜑𝐴 ∈ ℝ)
fourierdlem17.b (𝜑𝐵 ∈ ℝ)
fourierdlem17.altb (𝜑𝐴 < 𝐵)
fourierdlem17.l 𝐿 = (𝑥 ∈ (𝐴(,]𝐵) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥))
Assertion
Ref Expression
fourierdlem17 (𝜑𝐿:(𝐴(,]𝐵)⟶(𝐴[,]𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥
Allowed substitution hint:   𝐿(𝑥)

Proof of Theorem fourierdlem17
StepHypRef Expression
1 fourierdlem17.a . . . . 5 (𝜑𝐴 ∈ ℝ)
2 fourierdlem17.b . . . . 5 (𝜑𝐵 ∈ ℝ)
31leidd 10473 . . . . 5 (𝜑𝐴𝐴)
4 fourierdlem17.altb . . . . . 6 (𝜑𝐴 < 𝐵)
51, 2, 4ltled 10064 . . . . 5 (𝜑𝐴𝐵)
61, 2, 1, 3, 5eliccd 38573 . . . 4 (𝜑𝐴 ∈ (𝐴[,]𝐵))
76ad2antrr 758 . . 3 (((𝜑𝑥 ∈ (𝐴(,]𝐵)) ∧ 𝑥 = 𝐵) → 𝐴 ∈ (𝐴[,]𝐵))
8 iocssicc 12132 . . . . 5 (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵)
98sseli 3564 . . . 4 (𝑥 ∈ (𝐴(,]𝐵) → 𝑥 ∈ (𝐴[,]𝐵))
109ad2antlr 759 . . 3 (((𝜑𝑥 ∈ (𝐴(,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴[,]𝐵))
117, 10ifclda 4070 . 2 ((𝜑𝑥 ∈ (𝐴(,]𝐵)) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝐴[,]𝐵))
12 fourierdlem17.l . 2 𝐿 = (𝑥 ∈ (𝐴(,]𝐵) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥))
1311, 12fmptd 6292 1 (𝜑𝐿:(𝐴(,]𝐵)⟶(𝐴[,]𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1475  wcel 1977  ifcif 4036   class class class wbr 4583  cmpt 4643  wf 5800  (class class class)co 6549  cr 9814   < clt 9953  (,]cioc 12047  [,]cicc 12049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-ioc 12051  df-icc 12053
This theorem is referenced by:  fourierdlem79  39078  fourierdlem89  39088  fourierdlem90  39089  fourierdlem91  39090
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