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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvdmsscn | Structured version Visualization version GIF version |
Description: 𝑋 is a subset of ℂ. This statement is very often used when computing derivatives. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
dvdmsscn.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
dvdmsscn.x | ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) |
Ref | Expression |
---|---|
dvdmsscn | ⊢ (𝜑 → 𝑋 ⊆ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | restsspw 15915 | . . . 4 ⊢ ((TopOpen‘ℂfld) ↾t 𝑆) ⊆ 𝒫 𝑆 | |
2 | dvdmsscn.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆)) | |
3 | 1, 2 | sseldi 3566 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆) |
4 | elpwi 4117 | . . 3 ⊢ (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
6 | dvdmsscn.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
7 | recnprss 23474 | . . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
8 | 6, 7 | syl 17 | . 2 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
9 | 5, 8 | sstrd 3578 | 1 ⊢ (𝜑 → 𝑋 ⊆ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ⊆ wss 3540 𝒫 cpw 4108 {cpr 4127 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 ↾t crest 15904 TopOpenctopn 15905 ℂfldccnfld 19567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-rest 15906 |
This theorem is referenced by: dvxpaek 38830 etransclem17 39144 etransclem18 39145 etransclem20 39147 etransclem21 39148 etransclem22 39149 etransclem29 39156 etransclem31 39158 etransclem34 39161 etransclem43 39170 etransclem46 39173 |
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