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| Mirrors > Home > MPE Home > Th. List > cphdivcl | Structured version Visualization version GIF version | ||
| Description: The scalar field of a subcomplex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 11-Oct-2015.) |
| Ref | Expression |
|---|---|
| cphsca.f | β’ πΉ = (Scalarβπ) |
| cphsca.k | β’ πΎ = (BaseβπΉ) |
| Ref | Expression |
|---|---|
| cphdivcl | β’ ((π β βPreHil β§ (π΄ β πΎ β§ π΅ β πΎ β§ π΅ β 0)) β (π΄ / π΅) β πΎ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cphsca.f | . . . . . . 7 β’ πΉ = (Scalarβπ) | |
| 2 | cphsca.k | . . . . . . 7 β’ πΎ = (BaseβπΉ) | |
| 3 | 1, 2 | cphsubrg 25136 | . . . . . 6 β’ (π β βPreHil β πΎ β (SubRingββfld)) |
| 4 | 3 | adantr 479 | . . . . 5 β’ ((π β βPreHil β§ (π΄ β πΎ β§ π΅ β πΎ β§ π΅ β 0)) β πΎ β (SubRingββfld)) |
| 5 | cnfldbas 21297 | . . . . . 6 β’ β = (Baseββfld) | |
| 6 | 5 | subrgss 20525 | . . . . 5 β’ (πΎ β (SubRingββfld) β πΎ β β) |
| 7 | 4, 6 | syl 17 | . . . 4 β’ ((π β βPreHil β§ (π΄ β πΎ β§ π΅ β πΎ β§ π΅ β 0)) β πΎ β β) |
| 8 | simpr1 1191 | . . . 4 β’ ((π β βPreHil β§ (π΄ β πΎ β§ π΅ β πΎ β§ π΅ β 0)) β π΄ β πΎ) | |
| 9 | 7, 8 | sseldd 3983 | . . 3 β’ ((π β βPreHil β§ (π΄ β πΎ β§ π΅ β πΎ β§ π΅ β 0)) β π΄ β β) |
| 10 | simpr2 1192 | . . . 4 β’ ((π β βPreHil β§ (π΄ β πΎ β§ π΅ β πΎ β§ π΅ β 0)) β π΅ β πΎ) | |
| 11 | 7, 10 | sseldd 3983 | . . 3 β’ ((π β βPreHil β§ (π΄ β πΎ β§ π΅ β πΎ β§ π΅ β 0)) β π΅ β β) |
| 12 | simpr3 1193 | . . 3 β’ ((π β βPreHil β§ (π΄ β πΎ β§ π΅ β πΎ β§ π΅ β 0)) β π΅ β 0) | |
| 13 | 9, 11, 12 | divrecd 12033 | . 2 β’ ((π β βPreHil β§ (π΄ β πΎ β§ π΅ β πΎ β§ π΅ β 0)) β (π΄ / π΅) = (π΄ Β· (1 / π΅))) |
| 14 | 1, 2 | cphreccl 25137 | . . . 4 β’ ((π β βPreHil β§ π΅ β πΎ β§ π΅ β 0) β (1 / π΅) β πΎ) |
| 15 | 14 | 3adant3r1 1179 | . . 3 β’ ((π β βPreHil β§ (π΄ β πΎ β§ π΅ β πΎ β§ π΅ β 0)) β (1 / π΅) β πΎ) |
| 16 | cnfldmul 21301 | . . . 4 β’ Β· = (.rββfld) | |
| 17 | 16 | subrgmcl 20537 | . . 3 β’ ((πΎ β (SubRingββfld) β§ π΄ β πΎ β§ (1 / π΅) β πΎ) β (π΄ Β· (1 / π΅)) β πΎ) |
| 18 | 4, 8, 15, 17 | syl3anc 1368 | . 2 β’ ((π β βPreHil β§ (π΄ β πΎ β§ π΅ β πΎ β§ π΅ β 0)) β (π΄ Β· (1 / π΅)) β πΎ) |
| 19 | 13, 18 | eqeltrd 2829 | 1 β’ ((π β βPreHil β§ (π΄ β πΎ β§ π΅ β πΎ β§ π΅ β 0)) β (π΄ / π΅) β πΎ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2937 β wss 3949 βcfv 6553 (class class class)co 7426 βcc 11146 0cc0 11148 1c1 11149 Β· cmul 11153 / cdiv 11911 Basecbs 17189 Scalarcsca 17245 SubRingcsubrg 20520 βfldccnfld 21293 βPreHilccph 25122 |
| 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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-addf 11227 ax-mulf 11228 |
| 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 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8001 df-2nd 8002 df-tpos 8240 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-fz 13527 df-seq 14009 df-exp 14069 df-struct 17125 df-sets 17142 df-slot 17160 df-ndx 17172 df-base 17190 df-ress 17219 df-plusg 17255 df-mulr 17256 df-starv 17257 df-tset 17261 df-ple 17262 df-ds 17264 df-unif 17265 df-0g 17432 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-grp 18907 df-minusg 18908 df-subg 19092 df-cmn 19751 df-abl 19752 df-mgp 20089 df-rng 20107 df-ur 20136 df-ring 20189 df-cring 20190 df-oppr 20287 df-dvdsr 20310 df-unit 20311 df-invr 20341 df-dvr 20354 df-subrng 20497 df-subrg 20522 df-drng 20640 df-lvec 21002 df-cnfld 21294 df-phl 21572 df-cph 25124 |
| This theorem is referenced by: cphsqrtcl2 25142 pjthlem1 25393 |
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