Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhildrng | Structured version Visualization version GIF version |
Description: The star division ring for the final constructed Hilbert space is a division ring. (Contributed by NM, 20-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) |
Ref | Expression |
---|---|
hlhillvec.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hlhillvec.u | ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) |
hlhillvec.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hlhildrng.r | ⊢ 𝑅 = (Scalar‘𝑈) |
Ref | Expression |
---|---|
hlhildrng | ⊢ (𝜑 → 𝑅 ∈ DivRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhillvec.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | hlhillvec.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | eqid 2821 | . . . 4 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊) | |
4 | 2, 3 | erngdv 38144 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((EDRing‘𝐾)‘𝑊) ∈ DivRing) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝜑 → ((EDRing‘𝐾)‘𝑊) ∈ DivRing) |
6 | eqidd 2822 | . . 3 ⊢ (𝜑 → (Base‘((EDRing‘𝐾)‘𝑊)) = (Base‘((EDRing‘𝐾)‘𝑊))) | |
7 | hlhillvec.u | . . . 4 ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) | |
8 | hlhildrng.r | . . . 4 ⊢ 𝑅 = (Scalar‘𝑈) | |
9 | eqid 2821 | . . . 4 ⊢ (Base‘((EDRing‘𝐾)‘𝑊)) = (Base‘((EDRing‘𝐾)‘𝑊)) | |
10 | 2, 3, 7, 8, 1, 9 | hlhilsbase 39090 | . . 3 ⊢ (𝜑 → (Base‘((EDRing‘𝐾)‘𝑊)) = (Base‘𝑅)) |
11 | eqid 2821 | . . . . 5 ⊢ (+g‘((EDRing‘𝐾)‘𝑊)) = (+g‘((EDRing‘𝐾)‘𝑊)) | |
12 | 2, 3, 7, 8, 1, 11 | hlhilsplus 39091 | . . . 4 ⊢ (𝜑 → (+g‘((EDRing‘𝐾)‘𝑊)) = (+g‘𝑅)) |
13 | 12 | oveqdr 7184 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘((EDRing‘𝐾)‘𝑊)) ∧ 𝑦 ∈ (Base‘((EDRing‘𝐾)‘𝑊)))) → (𝑥(+g‘((EDRing‘𝐾)‘𝑊))𝑦) = (𝑥(+g‘𝑅)𝑦)) |
14 | eqid 2821 | . . . . 5 ⊢ (.r‘((EDRing‘𝐾)‘𝑊)) = (.r‘((EDRing‘𝐾)‘𝑊)) | |
15 | 2, 3, 7, 8, 1, 14 | hlhilsmul 39092 | . . . 4 ⊢ (𝜑 → (.r‘((EDRing‘𝐾)‘𝑊)) = (.r‘𝑅)) |
16 | 15 | oveqdr 7184 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘((EDRing‘𝐾)‘𝑊)) ∧ 𝑦 ∈ (Base‘((EDRing‘𝐾)‘𝑊)))) → (𝑥(.r‘((EDRing‘𝐾)‘𝑊))𝑦) = (𝑥(.r‘𝑅)𝑦)) |
17 | 6, 10, 13, 16 | drngpropd 19529 | . 2 ⊢ (𝜑 → (((EDRing‘𝐾)‘𝑊) ∈ DivRing ↔ 𝑅 ∈ DivRing)) |
18 | 5, 17 | mpbid 234 | 1 ⊢ (𝜑 → 𝑅 ∈ DivRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 Basecbs 16483 +gcplusg 16565 .rcmulr 16566 Scalarcsca 16568 DivRingcdr 19502 HLchlt 36501 LHypclh 37135 EDRingcedring 37904 HLHilchlh 39083 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-riotaBAD 36104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-undef 7939 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-0g 16715 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-llines 36649 df-lplanes 36650 df-lvols 36651 df-lines 36652 df-psubsp 36654 df-pmap 36655 df-padd 36947 df-lhyp 37139 df-laut 37140 df-ldil 37255 df-ltrn 37256 df-trl 37310 df-tendo 37906 df-edring 37908 df-hlhil 39084 |
This theorem is referenced by: hlhilsrnglem 39104 |
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