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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 11827 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐴) |
4 | fourierdlem17.altb | . . . . . 6 ⊢ (𝜑 → 𝐴 < 𝐵) | |
5 | 1, 2, 4 | ltled 11407 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
6 | 1, 2, 1, 3, 5 | eliccd 45457 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵)) |
7 | 6 | ad2antrr 726 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,]𝐵)) ∧ 𝑥 = 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) |
8 | iocssicc 13474 | . . . . 5 ⊢ (𝐴(,]𝐵) ⊆ (𝐴[,]𝐵) | |
9 | 8 | sseli 3991 | . . . 4 ⊢ (𝑥 ∈ (𝐴(,]𝐵) → 𝑥 ∈ (𝐴[,]𝐵)) |
10 | 9 | ad2antlr 727 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝐴(,]𝐵)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴[,]𝐵)) |
11 | 7, 10 | ifclda 4566 | . 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 7134 | 1 ⊢ (𝜑 → 𝐿:(𝐴(,]𝐵)⟶(𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ifcif 4531 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6559 (class class class)co 7431 ℝcr 11152 < clt 11293 (,]cioc 13385 [,]cicc 13387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-ioc 13389 df-icc 13391 |
This theorem is referenced by: fourierdlem79 46141 fourierdlem89 46151 fourierdlem90 46152 fourierdlem91 46153 |
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