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Mirrors > Home > MPE Home > Th. List > tcphphl | Structured version Visualization version GIF version |
Description: Augmentation of a subcomplex pre-Hilbert space with a norm does not affect whether it is still a pre-Hilbert space (because all the original components are the same). (Contributed by Mario Carneiro, 8-Oct-2015.) |
Ref | Expression |
---|---|
tcphval.n | ⊢ 𝐺 = (toℂPreHil‘𝑊) |
Ref | Expression |
---|---|
tcphphl | ⊢ (𝑊 ∈ PreHil ↔ 𝐺 ∈ PreHil) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2798 | . . 3 ⊢ (⊤ → (Base‘𝑊) = (Base‘𝑊)) | |
2 | tcphval.n | . . . . 5 ⊢ 𝐺 = (toℂPreHil‘𝑊) | |
3 | eqid 2797 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | 2, 3 | tcphbas 23509 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝐺) |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → (Base‘𝑊) = (Base‘𝐺)) |
6 | eqid 2797 | . . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
7 | 2, 6 | tchplusg 23510 | . . . . 5 ⊢ (+g‘𝑊) = (+g‘𝐺) |
8 | 7 | a1i 11 | . . . 4 ⊢ (⊤ → (+g‘𝑊) = (+g‘𝐺)) |
9 | 8 | oveqdr 7051 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐺)𝑦)) |
10 | eqidd 2798 | . . 3 ⊢ (⊤ → (Scalar‘𝑊) = (Scalar‘𝑊)) | |
11 | eqid 2797 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
12 | 2, 11 | tcphsca 23513 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝐺) |
13 | 12 | a1i 11 | . . 3 ⊢ (⊤ → (Scalar‘𝑊) = (Scalar‘𝐺)) |
14 | eqid 2797 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
15 | eqid 2797 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
16 | 2, 15 | tcphvsca 23514 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝐺) |
17 | 16 | a1i 11 | . . . 4 ⊢ (⊤ → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝐺)) |
18 | 17 | oveqdr 7051 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥( ·𝑠 ‘𝑊)𝑦) = (𝑥( ·𝑠 ‘𝐺)𝑦)) |
19 | eqid 2797 | . . . . . 6 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
20 | 2, 19 | tcphip 23515 | . . . . 5 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝐺) |
21 | 20 | a1i 11 | . . . 4 ⊢ (⊤ → (·𝑖‘𝑊) = (·𝑖‘𝐺)) |
22 | 21 | oveqdr 7051 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(·𝑖‘𝑊)𝑦) = (𝑥(·𝑖‘𝐺)𝑦)) |
23 | 1, 5, 9, 10, 13, 14, 18, 22 | phlpropd 20485 | . 2 ⊢ (⊤ → (𝑊 ∈ PreHil ↔ 𝐺 ∈ PreHil)) |
24 | 23 | mptru 1532 | 1 ⊢ (𝑊 ∈ PreHil ↔ 𝐺 ∈ PreHil) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1525 ⊤wtru 1526 ∈ wcel 2083 ‘cfv 6232 Basecbs 16316 +gcplusg 16398 Scalarcsca 16401 ·𝑠 cvsca 16402 ·𝑖cip 16403 PreHilcphl 20454 toℂPreHilctcph 23458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-er 8146 df-map 8265 df-en 8365 df-dom 8366 df-sdom 8367 df-sup 8759 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-z 11836 df-dec 11953 df-uz 12098 df-rp 12244 df-seq 13224 df-exp 13284 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-sca 16414 df-vsca 16415 df-ip 16416 df-tset 16417 df-ds 16420 df-0g 16548 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-mhm 17778 df-grp 17868 df-ghm 18101 df-mgp 18934 df-ur 18946 df-ring 18993 df-lmod 19330 df-lmhm 19488 df-lvec 19569 df-sra 19638 df-rgmod 19639 df-phl 20456 df-tng 22881 df-tcph 23460 |
This theorem is referenced by: tcphcph 23527 |
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