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Mirrors > Home > MPE Home > Th. List > dvrcn | Structured version Visualization version GIF version |
Description: The division function is continuous in a topological field. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
dvrcn.j | ⊢ 𝐽 = (TopOpen‘𝑅) |
dvrcn.d | ⊢ / = (/r‘𝑅) |
dvrcn.u | ⊢ 𝑈 = (Unit‘𝑅) |
Ref | Expression |
---|---|
dvrcn | ⊢ (𝑅 ∈ TopDRing → / ∈ ((𝐽 ×t (𝐽 ↾t 𝑈)) Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2610 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
3 | dvrcn.u | . . 3 ⊢ 𝑈 = (Unit‘𝑅) | |
4 | eqid 2610 | . . 3 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
5 | dvrcn.d | . . 3 ⊢ / = (/r‘𝑅) | |
6 | 1, 2, 3, 4, 5 | dvrfval 18507 | . 2 ⊢ / = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ 𝑈 ↦ (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦))) |
7 | dvrcn.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝑅) | |
8 | tdrgtrg 21786 | . . 3 ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing) | |
9 | tdrgtps 21790 | . . . 4 ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopSp) | |
10 | 1, 7 | istps 20551 | . . . 4 ⊢ (𝑅 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑅))) |
11 | 9, 10 | sylib 207 | . . 3 ⊢ (𝑅 ∈ TopDRing → 𝐽 ∈ (TopOn‘(Base‘𝑅))) |
12 | 1, 3 | unitss 18483 | . . . 4 ⊢ 𝑈 ⊆ (Base‘𝑅) |
13 | resttopon 20775 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘(Base‘𝑅)) ∧ 𝑈 ⊆ (Base‘𝑅)) → (𝐽 ↾t 𝑈) ∈ (TopOn‘𝑈)) | |
14 | 11, 12, 13 | sylancl 693 | . . 3 ⊢ (𝑅 ∈ TopDRing → (𝐽 ↾t 𝑈) ∈ (TopOn‘𝑈)) |
15 | 11, 14 | cnmpt1st 21281 | . . 3 ⊢ (𝑅 ∈ TopDRing → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ 𝑈 ↦ 𝑥) ∈ ((𝐽 ×t (𝐽 ↾t 𝑈)) Cn 𝐽)) |
16 | 11, 14 | cnmpt2nd 21282 | . . . 4 ⊢ (𝑅 ∈ TopDRing → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ 𝑈 ↦ 𝑦) ∈ ((𝐽 ×t (𝐽 ↾t 𝑈)) Cn (𝐽 ↾t 𝑈))) |
17 | 7, 4, 3 | invrcn 21794 | . . . 4 ⊢ (𝑅 ∈ TopDRing → (invr‘𝑅) ∈ ((𝐽 ↾t 𝑈) Cn 𝐽)) |
18 | 11, 14, 16, 17 | cnmpt21f 21285 | . . 3 ⊢ (𝑅 ∈ TopDRing → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ 𝑈 ↦ ((invr‘𝑅)‘𝑦)) ∈ ((𝐽 ×t (𝐽 ↾t 𝑈)) Cn 𝐽)) |
19 | 7, 2, 8, 11, 14, 15, 18 | cnmpt2mulr 21796 | . 2 ⊢ (𝑅 ∈ TopDRing → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ 𝑈 ↦ (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦))) ∈ ((𝐽 ×t (𝐽 ↾t 𝑈)) Cn 𝐽)) |
20 | 6, 19 | syl5eqel 2692 | 1 ⊢ (𝑅 ∈ TopDRing → / ∈ ((𝐽 ×t (𝐽 ↾t 𝑈)) Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 Basecbs 15695 .rcmulr 15769 ↾t crest 15904 TopOpenctopn 15905 Unitcui 18462 invrcinvr 18494 /rcdvr 18505 TopOnctopon 20518 TopSpctps 20519 Cn ccn 20838 ×t ctx 21173 TopDRingctdrg 21770 |
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-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-nel 2783 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-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 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-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-tset 15787 df-rest 15906 df-topn 15907 df-topgen 15927 df-plusf 17064 df-minusg 17249 df-mgp 18313 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cn 20841 df-tx 21175 df-tmd 21686 df-tgp 21687 df-trg 21773 df-tdrg 21774 |
This theorem is referenced by: (None) |
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