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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 11208 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐴) |
4 | fourierdlem17.altb | . . . . . 6 ⊢ (𝜑 → 𝐴 < 𝐵) | |
5 | 1, 2, 4 | ltled 10790 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
6 | 1, 2, 1, 3, 5 | eliccd 41786 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
7 | 6 | ad2antrr 724 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,]𝐵)) ∧ 𝑥 = 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) |
8 | iocssicc 12828 | . . . . 5 ⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵) | |
9 | 8 | sseli 3965 | . . . 4 ⊢ (𝑥 ∈ (𝐴(,]𝐵) → 𝑥 ∈ (𝐴[,]𝐵)) |
10 | 9 | ad2antlr 725 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴[,]𝐵)) |
11 | 7, 10 | ifclda 4503 | . 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 6880 | 1 ⊢ (𝜑 → 𝐿:(𝐴(,]𝐵)⟶(𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ifcif 4469 class class class wbr 5068 ↦ cmpt 5148 ⟶wf 6353 (class class class)co 7158 ℝcr 10538 < clt 10677 (,]cioc 12742 [,]cicc 12744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-ioc 12746 df-icc 12748 |
This theorem is referenced by: fourierdlem79 42477 fourierdlem89 42487 fourierdlem90 42488 fourierdlem91 42489 |
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