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Mirrors > Home > MPE Home > Th. List > invrcn | Structured version Visualization version GIF version |
Description: The multiplicative inverse function is a continuous function from the unit group (that is, the nonzero numbers) to the field. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
mulrcn.j | ⊢ 𝐽 = (TopOpen‘𝑅) |
invrcn.i | ⊢ 𝐼 = (invr‘𝑅) |
invrcn.u | ⊢ 𝑈 = (Unit‘𝑅) |
Ref | Expression |
---|---|
invrcn | ⊢ (𝑅 ∈ TopDRing → 𝐼 ∈ ((𝐽 ↾t 𝑈) Cn 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tdrgtps 24099 | . . 3 ⊢ (𝑅 ∈ TopDRing → 𝑅 ∈ TopSp) | |
2 | mulrcn.j | . . . 4 ⊢ 𝐽 = (TopOpen‘𝑅) | |
3 | 2 | tpstop 22857 | . . 3 ⊢ (𝑅 ∈ TopSp → 𝐽 ∈ Top) |
4 | cnrest2r 23209 | . . 3 ⊢ (𝐽 ∈ Top → ((𝐽 ↾t 𝑈) Cn (𝐽 ↾t 𝑈)) ⊆ ((𝐽 ↾t 𝑈) Cn 𝐽)) | |
5 | 1, 3, 4 | 3syl 18 | . 2 ⊢ (𝑅 ∈ TopDRing → ((𝐽 ↾t 𝑈) Cn (𝐽 ↾t 𝑈)) ⊆ ((𝐽 ↾t 𝑈) Cn 𝐽)) |
6 | invrcn.i | . . 3 ⊢ 𝐼 = (invr‘𝑅) | |
7 | invrcn.u | . . 3 ⊢ 𝑈 = (Unit‘𝑅) | |
8 | 2, 6, 7 | invrcn2 24102 | . 2 ⊢ (𝑅 ∈ TopDRing → 𝐼 ∈ ((𝐽 ↾t 𝑈) Cn (𝐽 ↾t 𝑈))) |
9 | 5, 8 | sseldd 3981 | 1 ⊢ (𝑅 ∈ TopDRing → 𝐼 ∈ ((𝐽 ↾t 𝑈) Cn 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3947 ‘cfv 6551 (class class class)co 7424 ↾t crest 17407 TopOpenctopn 17408 Unitcui 20299 invrcinvr 20331 Topctop 22813 TopSpctps 22852 Cn ccn 23146 TopDRingctdrg 24079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fi 9440 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-tset 17257 df-rest 17409 df-topn 17410 df-topgen 17430 df-minusg 18899 df-mgp 20080 df-invr 20332 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cn 23149 df-tmd 23994 df-tgp 23995 df-trg 24082 df-tdrg 24083 |
This theorem is referenced by: dvrcn 24106 |
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