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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 2731 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2731 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
3 | dvrcn.u | . . 3 ⊢ 𝑈 = (Unit‘𝑅) | |
4 | eqid 2731 | . . 3 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
5 | dvrcn.d | . . 3 ⊢ / = (/r‘𝑅) | |
6 | 1, 2, 3, 4, 5 | dvrfval 20300 | . 2 ⊢ / = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ 𝑈 ↦ (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦))) |
7 | dvrcn.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝑅) | |
8 | tdrgtrg 23997 | . . 3 ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopRing) | |
9 | tdrgtps 24001 | . . . 4 ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopSp) | |
10 | 1, 7 | istps 22756 | . . . 4 ⊢ (𝑅 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑅))) |
11 | 9, 10 | sylib 217 | . . 3 ⊢ (𝑅 ∈ TopDRing → 𝐽 ∈ (TopOn‘(Base‘𝑅))) |
12 | 1, 3 | unitss 20274 | . . . 4 ⊢ 𝑈 ⊆ (Base‘𝑅) |
13 | resttopon 22985 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘(Base‘𝑅)) ∧ 𝑈 ⊆ (Base‘𝑅)) → (𝐽 ↾t 𝑈) ∈ (TopOn‘𝑈)) | |
14 | 11, 12, 13 | sylancl 585 | . . 3 ⊢ (𝑅 ∈ TopDRing → (𝐽 ↾t 𝑈) ∈ (TopOn‘𝑈)) |
15 | 11, 14 | cnmpt1st 23492 | . . 3 ⊢ (𝑅 ∈ TopDRing → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ 𝑈 ↦ 𝑥) ∈ ((𝐽 ×t (𝐽 ↾t 𝑈)) Cn 𝐽)) |
16 | 11, 14 | cnmpt2nd 23493 | . . . 4 ⊢ (𝑅 ∈ TopDRing → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ 𝑈 ↦ 𝑦) ∈ ((𝐽 ×t (𝐽 ↾t 𝑈)) Cn (𝐽 ↾t 𝑈))) |
17 | 7, 4, 3 | invrcn 24005 | . . . 4 ⊢ (𝑅 ∈ TopDRing → (invr‘𝑅) ∈ ((𝐽 ↾t 𝑈) Cn 𝐽)) |
18 | 11, 14, 16, 17 | cnmpt21f 23496 | . . 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 24007 | . 2 ⊢ (𝑅 ∈ TopDRing → (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ 𝑈 ↦ (𝑥(.r‘𝑅)((invr‘𝑅)‘𝑦))) ∈ ((𝐽 ×t (𝐽 ↾t 𝑈)) Cn 𝐽)) |
20 | 6, 19 | eqeltrid 2836 | 1 ⊢ (𝑅 ∈ TopDRing → / ∈ ((𝐽 ×t (𝐽 ↾t 𝑈)) Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ⊆ wss 3948 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 Basecbs 17151 .rcmulr 17205 ↾t crest 17373 TopOpenctopn 17374 Unitcui 20253 invrcinvr 20285 /rcdvr 20298 TopOnctopon 22732 TopSpctps 22754 Cn ccn 23048 ×t ctx 23384 TopDRingctdrg 23981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fi 9412 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-tset 17223 df-rest 17375 df-topn 17376 df-topgen 17396 df-plusf 18570 df-minusg 18865 df-mgp 20036 df-dvdsr 20255 df-unit 20256 df-invr 20286 df-dvr 20299 df-top 22716 df-topon 22733 df-topsp 22755 df-bases 22769 df-cn 23051 df-tx 23386 df-tmd 23896 df-tgp 23897 df-trg 23984 df-tdrg 23985 |
This theorem is referenced by: (None) |
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