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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 2727 | . . 3 ⊢ (⊤ → (Base‘𝑊) = (Base‘𝑊)) | |
2 | tcphval.n | . . . . 5 ⊢ 𝐺 = (toℂPreHil‘𝑊) | |
3 | eqid 2726 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | 2, 3 | tcphbas 25238 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝐺) |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → (Base‘𝑊) = (Base‘𝐺)) |
6 | eqid 2726 | . . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
7 | 2, 6 | tchplusg 25239 | . . . . 5 ⊢ (+g‘𝑊) = (+g‘𝐺) |
8 | 7 | a1i 11 | . . . 4 ⊢ (⊤ → (+g‘𝑊) = (+g‘𝐺)) |
9 | 8 | oveqdr 7452 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐺)𝑦)) |
10 | eqidd 2727 | . . 3 ⊢ (⊤ → (Scalar‘𝑊) = (Scalar‘𝑊)) | |
11 | eqid 2726 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
12 | 2, 11 | tcphsca 25242 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝐺) |
13 | 12 | a1i 11 | . . 3 ⊢ (⊤ → (Scalar‘𝑊) = (Scalar‘𝐺)) |
14 | eqid 2726 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
15 | eqid 2726 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
16 | 2, 15 | tcphvsca 25243 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝐺) |
17 | 16 | a1i 11 | . . . 4 ⊢ (⊤ → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝐺)) |
18 | 17 | oveqdr 7452 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥( ·𝑠 ‘𝑊)𝑦) = (𝑥( ·𝑠 ‘𝐺)𝑦)) |
19 | eqid 2726 | . . . . . 6 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
20 | 2, 19 | tcphip 25244 | . . . . 5 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝐺) |
21 | 20 | a1i 11 | . . . 4 ⊢ (⊤ → (·𝑖‘𝑊) = (·𝑖‘𝐺)) |
22 | 21 | oveqdr 7452 | . . 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 21651 | . 2 ⊢ (⊤ → (𝑊 ∈ PreHil ↔ 𝐺 ∈ PreHil)) |
24 | 23 | mptru 1541 | 1 ⊢ (𝑊 ∈ PreHil ↔ 𝐺 ∈ PreHil) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 ‘cfv 6554 Basecbs 17213 +gcplusg 17266 Scalarcsca 17269 ·𝑠 cvsca 17270 ·𝑖cip 17271 PreHilcphl 21620 toℂPreHilctcph 25186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 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 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-sup 9485 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-rp 13029 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ds 17288 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-mhm 18773 df-grp 18931 df-ghm 19207 df-mgp 20118 df-ur 20165 df-ring 20218 df-lmod 20838 df-lmhm 21000 df-lvec 21081 df-sra 21151 df-rgmod 21152 df-phl 21622 df-tng 24584 df-tcph 25188 |
This theorem is referenced by: tcphcph 25256 |
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