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Mirrors > Home > MPE Home > Th. List > phclm | Structured version Visualization version GIF version |
Description: A pre-Hilbert space whose field of scalars is a restriction of the field of complex numbers is a subcomplex module. TODO: redundant hypotheses. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
tcphval.n | ⊢ 𝐺 = (toℂPreHil‘𝑊) |
tcphcph.v | ⊢ 𝑉 = (Base‘𝑊) |
tcphcph.f | ⊢ 𝐹 = (Scalar‘𝑊) |
tcphcph.1 | ⊢ (𝜑 → 𝑊 ∈ PreHil) |
tcphcph.2 | ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐾)) |
Ref | Expression |
---|---|
phclm | ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tcphcph.1 | . . 3 ⊢ (𝜑 → 𝑊 ∈ PreHil) | |
2 | phllmod 20769 | . . 3 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝑊 ∈ LMod) |
4 | eqid 2820 | . . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
5 | tcphcph.2 | . . . 4 ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐾)) | |
6 | phllvec 20768 | . . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LVec) | |
7 | 1, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LVec) |
8 | tcphcph.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
9 | 8 | lvecdrng 19872 | . . . . 5 ⊢ (𝑊 ∈ LVec → 𝐹 ∈ DivRing) |
10 | 7, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ DivRing) |
11 | 4, 5, 10 | cphsubrglem 23776 | . . 3 ⊢ (𝜑 → (𝐹 = (ℂfld ↾s (Base‘𝐹)) ∧ (Base‘𝐹) = (𝐾 ∩ ℂ) ∧ (Base‘𝐹) ∈ (SubRing‘ℂfld))) |
12 | 11 | simp1d 1137 | . 2 ⊢ (𝜑 → 𝐹 = (ℂfld ↾s (Base‘𝐹))) |
13 | 11 | simp3d 1139 | . 2 ⊢ (𝜑 → (Base‘𝐹) ∈ (SubRing‘ℂfld)) |
14 | 8, 4 | isclm 23663 | . 2 ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s (Base‘𝐹)) ∧ (Base‘𝐹) ∈ (SubRing‘ℂfld))) |
15 | 3, 12, 13, 14 | syl3anbrc 1338 | 1 ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ∩ cin 3928 ‘cfv 6348 (class class class)co 7149 ℂcc 10528 Basecbs 16478 ↾s cress 16479 Scalarcsca 16563 DivRingcdr 19497 SubRingcsubrg 19526 LModclmod 19629 LVecclvec 19869 ℂfldccnfld 20540 PreHilcphl 20763 ℂModcclm 23661 toℂPreHilctcph 23766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-addf 10609 ax-mulf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-tpos 7885 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-fz 12890 df-seq 13367 df-exp 13427 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-starv 16575 df-tset 16579 df-ple 16580 df-ds 16582 df-unif 16583 df-0g 16710 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-grp 18101 df-subg 18271 df-cmn 18903 df-mgp 19235 df-ur 19247 df-ring 19294 df-cring 19295 df-oppr 19368 df-dvdsr 19386 df-unit 19387 df-drng 19499 df-subrg 19528 df-lvec 19870 df-cnfld 20541 df-phl 20765 df-clm 23662 |
This theorem is referenced by: tcphcphlem3 23831 ipcau2 23832 tcphcphlem1 23833 tcphcphlem2 23834 tcphcph 23835 |
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