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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rmulccn | Structured version Visualization version GIF version |
Description: Multiplication by a real constant is a continuous function. (Contributed by Thierry Arnoux, 23-May-2017.) Avoid ax-mulf 11266. (Revised by GG, 16-Mar-2025.) |
Ref | Expression |
---|---|
rmulccn.1 | ⊢ 𝐽 = (topGen‘ran (,)) |
rmulccn.2 | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
Ref | Expression |
---|---|
rmulccn | ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ (𝑥 · 𝐶)) ∈ (𝐽 Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2740 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
2 | 1 | cnfldtopon 24826 | . . . . . 6 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝜑 → (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)) |
4 | 3 | cnmptid 23692 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝑥) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
5 | rmulccn.2 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
6 | 5 | recnd 11320 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
7 | 3, 3, 6 | cnmptc 23693 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ 𝐶) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
8 | 1 | mpomulcn 24912 | . . . . . 6 ⊢ (𝑦 ∈ ℂ, 𝑧 ∈ ℂ ↦ (𝑦 · 𝑧)) ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld)) |
9 | 8 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑦 ∈ ℂ, 𝑧 ∈ ℂ ↦ (𝑦 · 𝑧)) ∈ (((TopOpen‘ℂfld) ×t (TopOpen‘ℂfld)) Cn (TopOpen‘ℂfld))) |
10 | oveq12 7459 | . . . . 5 ⊢ ((𝑦 = 𝑥 ∧ 𝑧 = 𝐶) → (𝑦 · 𝑧) = (𝑥 · 𝐶)) | |
11 | 3, 4, 7, 3, 3, 9, 10 | cnmpt12 23698 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld))) |
12 | ax-resscn 11243 | . . . 4 ⊢ ℝ ⊆ ℂ | |
13 | unicntop 24829 | . . . . 5 ⊢ ℂ = ∪ (TopOpen‘ℂfld) | |
14 | 13 | cnrest 23316 | . . . 4 ⊢ (((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ∈ ((TopOpen‘ℂfld) Cn (TopOpen‘ℂfld)) ∧ ℝ ⊆ ℂ) → ((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ) ∈ (((TopOpen‘ℂfld) ↾t ℝ) Cn (TopOpen‘ℂfld))) |
15 | 11, 12, 14 | sylancl 585 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ) ∈ (((TopOpen‘ℂfld) ↾t ℝ) Cn (TopOpen‘ℂfld))) |
16 | simpr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ) | |
17 | 6 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐶 ∈ ℂ) |
18 | 16, 17 | mulcld 11312 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑥 · 𝐶) ∈ ℂ) |
19 | 18 | ralrimiva 3152 | . . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ ℂ (𝑥 · 𝐶) ∈ ℂ) |
20 | eqid 2740 | . . . . . . . 8 ⊢ (𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) = (𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) | |
21 | 20 | fnmpt 6722 | . . . . . . 7 ⊢ (∀𝑥 ∈ ℂ (𝑥 · 𝐶) ∈ ℂ → (𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) Fn ℂ) |
22 | 19, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) Fn ℂ) |
23 | 12 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ) |
24 | 22, 23 | fnssresd 6706 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ) Fn ℝ) |
25 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → 𝑤 ∈ ℝ) | |
26 | oveq1 7457 | . . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑥 · 𝐶) = (𝑤 · 𝐶)) | |
27 | resmpt 6068 | . . . . . . . . . 10 ⊢ (ℝ ⊆ ℂ → ((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ) = (𝑥 ∈ ℝ ↦ (𝑥 · 𝐶))) | |
28 | 12, 27 | ax-mp 5 | . . . . . . . . 9 ⊢ ((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ) = (𝑥 ∈ ℝ ↦ (𝑥 · 𝐶)) |
29 | ovex 7483 | . . . . . . . . 9 ⊢ (𝑤 · 𝐶) ∈ V | |
30 | 26, 28, 29 | fvmpt 7031 | . . . . . . . 8 ⊢ (𝑤 ∈ ℝ → (((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ)‘𝑤) = (𝑤 · 𝐶)) |
31 | 25, 30 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ)‘𝑤) = (𝑤 · 𝐶)) |
32 | 5 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → 𝐶 ∈ ℝ) |
33 | 25, 32 | remulcld 11322 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (𝑤 · 𝐶) ∈ ℝ) |
34 | 31, 33 | eqeltrd 2844 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ)‘𝑤) ∈ ℝ) |
35 | 34 | ralrimiva 3152 | . . . . 5 ⊢ (𝜑 → ∀𝑤 ∈ ℝ (((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ)‘𝑤) ∈ ℝ) |
36 | fnfvrnss 7157 | . . . . 5 ⊢ ((((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ) Fn ℝ ∧ ∀𝑤 ∈ ℝ (((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ)‘𝑤) ∈ ℝ) → ran ((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ) ⊆ ℝ) | |
37 | 24, 35, 36 | syl2anc 583 | . . . 4 ⊢ (𝜑 → ran ((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ) ⊆ ℝ) |
38 | cnrest2 23317 | . . . 4 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ ran ((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ) ⊆ ℝ ∧ ℝ ⊆ ℂ) → (((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ) ∈ (((TopOpen‘ℂfld) ↾t ℝ) Cn (TopOpen‘ℂfld)) ↔ ((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ) ∈ (((TopOpen‘ℂfld) ↾t ℝ) Cn ((TopOpen‘ℂfld) ↾t ℝ)))) | |
39 | 2, 37, 23, 38 | mp3an2i 1466 | . . 3 ⊢ (𝜑 → (((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ) ∈ (((TopOpen‘ℂfld) ↾t ℝ) Cn (TopOpen‘ℂfld)) ↔ ((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ) ∈ (((TopOpen‘ℂfld) ↾t ℝ) Cn ((TopOpen‘ℂfld) ↾t ℝ)))) |
40 | 15, 39 | mpbid 232 | . 2 ⊢ (𝜑 → ((𝑥 ∈ ℂ ↦ (𝑥 · 𝐶)) ↾ ℝ) ∈ (((TopOpen‘ℂfld) ↾t ℝ) Cn ((TopOpen‘ℂfld) ↾t ℝ))) |
41 | rmulccn.1 | . . . . 5 ⊢ 𝐽 = (topGen‘ran (,)) | |
42 | 1 | tgioo2 24846 | . . . . 5 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
43 | 41, 42 | eqtri 2768 | . . . 4 ⊢ 𝐽 = ((TopOpen‘ℂfld) ↾t ℝ) |
44 | 43, 43 | oveq12i 7462 | . . 3 ⊢ (𝐽 Cn 𝐽) = (((TopOpen‘ℂfld) ↾t ℝ) Cn ((TopOpen‘ℂfld) ↾t ℝ)) |
45 | 44 | eqcomi 2749 | . 2 ⊢ (((TopOpen‘ℂfld) ↾t ℝ) Cn ((TopOpen‘ℂfld) ↾t ℝ)) = (𝐽 Cn 𝐽) |
46 | 40, 28, 45 | 3eltr3g 2860 | 1 ⊢ (𝜑 → (𝑥 ∈ ℝ ↦ (𝑥 · 𝐶)) ∈ (𝐽 Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ⊆ wss 3976 ↦ cmpt 5249 ran crn 5701 ↾ cres 5702 Fn wfn 6570 ‘cfv 6575 (class class class)co 7450 ∈ cmpo 7452 ℂcc 11184 ℝcr 11185 · cmul 11191 (,)cioo 13409 ↾t crest 17482 TopOpenctopn 17483 topGenctg 17499 ℂfldccnfld 21389 TopOnctopon 22939 Cn ccn 23255 ×t ctx 23591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-pre-sup 11264 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-isom 6584 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-of 7716 df-om 7906 df-1st 8032 df-2nd 8033 df-supp 8204 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-2o 8525 df-er 8765 df-map 8888 df-ixp 8958 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-fsupp 9434 df-fi 9482 df-sup 9513 df-inf 9514 df-oi 9581 df-card 10010 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-div 11950 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-7 12363 df-8 12364 df-9 12365 df-n0 12556 df-z 12642 df-dec 12761 df-uz 12906 df-q 13016 df-rp 13060 df-xneg 13177 df-xadd 13178 df-xmul 13179 df-ioo 13413 df-icc 13416 df-fz 13570 df-fzo 13714 df-seq 14055 df-exp 14115 df-hash 14382 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-struct 17196 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-ress 17290 df-plusg 17326 df-mulr 17327 df-starv 17328 df-sca 17329 df-vsca 17330 df-ip 17331 df-tset 17332 df-ple 17333 df-ds 17335 df-unif 17336 df-hom 17337 df-cco 17338 df-rest 17484 df-topn 17485 df-0g 17503 df-gsum 17504 df-topgen 17505 df-pt 17506 df-prds 17509 df-xrs 17564 df-qtop 17569 df-imas 17570 df-xps 17572 df-mre 17646 df-mrc 17647 df-acs 17649 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-submnd 18821 df-mulg 19110 df-cntz 19359 df-cmn 19826 df-psmet 21381 df-xmet 21382 df-met 21383 df-bl 21384 df-mopn 21385 df-cnfld 21390 df-top 22923 df-topon 22940 df-topsp 22962 df-bases 22976 df-cn 23258 df-cnp 23259 df-tx 23593 df-hmeo 23786 df-xms 24353 df-ms 24354 df-tms 24355 |
This theorem is referenced by: rrvmulc 34420 |
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