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Mirrors > Home > MPE Home > Th. List > Mathboxes > fourierdlem17 | Structured version Visualization version GIF version |
Description: The defined 𝐿 is actually a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
fourierdlem17.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
fourierdlem17.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
fourierdlem17.altb | ⊢ (𝜑 → 𝐴 < 𝐵) |
fourierdlem17.l | ⊢ 𝐿 = (𝑥 ∈ (𝐴(,]𝐵) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥)) |
Ref | Expression |
---|---|
fourierdlem17 | ⊢ (𝜑 → 𝐿:(𝐴(,]𝐵)⟶(𝐴[,]𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fourierdlem17.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | fourierdlem17.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | 1 | leidd 11529 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐴) |
4 | fourierdlem17.altb | . . . . . 6 ⊢ (𝜑 → 𝐴 < 𝐵) | |
5 | 1, 2, 4 | ltled 11111 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
6 | 1, 2, 1, 3, 5 | eliccd 43001 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
7 | 6 | ad2antrr 723 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,]𝐵)) ∧ 𝑥 = 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) |
8 | iocssicc 13157 | . . . . 5 ⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵) | |
9 | 8 | sseli 3917 | . . . 4 ⊢ (𝑥 ∈ (𝐴(,]𝐵) → 𝑥 ∈ (𝐴[,]𝐵)) |
10 | 9 | ad2antlr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴[,]𝐵)) |
11 | 7, 10 | ifclda 4495 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,]𝐵)) → if(𝑥 = 𝐵, 𝐴, 𝑥) ∈ (𝐴[,]𝐵)) |
12 | fourierdlem17.l | . 2 ⊢ 𝐿 = (𝑥 ∈ (𝐴(,]𝐵) ↦ if(𝑥 = 𝐵, 𝐴, 𝑥)) | |
13 | 11, 12 | fmptd 6981 | 1 ⊢ (𝜑 → 𝐿:(𝐴(,]𝐵)⟶(𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ifcif 4460 class class class wbr 5074 ↦ cmpt 5157 ⟶wf 6423 (class class class)co 7268 ℝcr 10858 < clt 10997 (,]cioc 13068 [,]cicc 13070 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-cnex 10915 ax-resscn 10916 ax-pre-lttri 10933 ax-pre-lttrn 10934 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5485 df-po 5499 df-so 5500 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-ov 7271 df-oprab 7272 df-mpo 7273 df-er 8486 df-en 8722 df-dom 8723 df-sdom 8724 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-ioc 13072 df-icc 13074 |
This theorem is referenced by: fourierdlem79 43685 fourierdlem89 43695 fourierdlem90 43696 fourierdlem91 43697 |
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