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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 11209 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐴) |
4 | fourierdlem17.altb | . . . . . 6 ⊢ (𝜑 → 𝐴 < 𝐵) | |
5 | 1, 2, 4 | ltled 10791 | . . . . 5 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
6 | 1, 2, 1, 3, 5 | eliccd 41785 | . . . 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 3966 | . . . 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 4504 | . 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 6881 | 1 ⊢ (𝜑 → 𝐿:(𝐴(,]𝐵)⟶(𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ifcif 4470 class class class wbr 5069 ↦ cmpt 5149 ⟶wf 6354 (class class class)co 7159 ℝcr 10539 < clt 10678 (,]cioc 12742 [,]cicc 12744 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-pre-lttri 10614 ax-pre-lttrn 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-ioc 12746 df-icc 12748 |
This theorem is referenced by: fourierdlem79 42477 fourierdlem89 42487 fourierdlem90 42488 fourierdlem91 42489 |
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