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Mathbox for Glauco Siliprandi |
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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 11784 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐴) |
4 | fourierdlem17.altb | . . . . . 6 ⊢ (𝜑 → 𝐴 < 𝐵) | |
5 | 1, 2, 4 | ltled 11366 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
6 | 1, 2, 1, 3, 5 | eliccd 44515 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
7 | 6 | ad2antrr 722 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,]𝐵)) ∧ 𝑥 = 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) |
8 | iocssicc 13418 | . . . . 5 ⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵) | |
9 | 8 | sseli 3977 | . . . 4 ⊢ (𝑥 ∈ (𝐴(,]𝐵) → 𝑥 ∈ (𝐴[,]𝐵)) |
10 | 9 | ad2antlr 723 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴[,]𝐵)) |
11 | 7, 10 | ifclda 4562 | . 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 7114 | 1 ⊢ (𝜑 → 𝐿:(𝐴(,]𝐵)⟶(𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ifcif 4527 class class class wbr 5147 ↦ cmpt 5230 ⟶wf 6538 (class class class)co 7411 ℝcr 11111 < clt 11252 (,]cioc 13329 [,]cicc 13331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 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-ov 7414 df-oprab 7415 df-mpo 7416 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-ioc 13333 df-icc 13335 |
This theorem is referenced by: fourierdlem79 45199 fourierdlem89 45209 fourierdlem90 45210 fourierdlem91 45211 |
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