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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 25063 | . . . . . 6 β’ (π β βPreHil β πΎ β (SubRingββfld)) |
| 4 | 3 | adantr 480 | . . . . 5 β’ ((π β βPreHil β§ (π΄ β πΎ β§ π΅ β πΎ β§ π΅ β 0)) β πΎ β (SubRingββfld)) |
| 5 | cnfldbas 21244 | . . . . . 6 β’ β = (Baseββfld) | |
| 6 | 5 | subrgss 20474 | . . . . 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 3978 | . . 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 3978 | . . 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 11997 | . 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 25064 | . . . 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 21248 | . . . 4 β’ Β· = (.rββfld) | |
| 17 | 16 | subrgmcl 20486 | . . 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 2827 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 β wss 3943 βcfv 6537 (class class class)co 7405 βcc 11110 0cc0 11112 1c1 11113 Β· cmul 11117 / cdiv 11875 Basecbs 17153 Scalarcsca 17209 SubRingcsubrg 20469 βfldccnfld 21240 βPreHilccph 25049 |
| 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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-addf 11191 ax-mulf 11192 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-seq 13973 df-exp 14033 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-0g 17396 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-grp 18866 df-minusg 18867 df-subg 19050 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-cring 20141 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-subrng 20446 df-subrg 20471 df-drng 20589 df-lvec 20951 df-cnfld 21241 df-phl 21519 df-cph 25051 |
| This theorem is referenced by: cphsqrtcl2 25069 pjthlem1 25320 |
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