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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 2735 | . . 3 ⊢ (⊤ → (Base‘𝑊) = (Base‘𝑊)) | |
2 | tcphval.n | . . . . 5 ⊢ 𝐺 = (toℂPreHil‘𝑊) | |
3 | eqid 2734 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | 2, 3 | tcphbas 25265 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝐺) |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → (Base‘𝑊) = (Base‘𝐺)) |
6 | eqid 2734 | . . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
7 | 2, 6 | tchplusg 25266 | . . . . 5 ⊢ (+g‘𝑊) = (+g‘𝐺) |
8 | 7 | a1i 11 | . . . 4 ⊢ (⊤ → (+g‘𝑊) = (+g‘𝐺)) |
9 | 8 | oveqdr 7473 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐺)𝑦)) |
10 | eqidd 2735 | . . 3 ⊢ (⊤ → (Scalar‘𝑊) = (Scalar‘𝑊)) | |
11 | eqid 2734 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
12 | 2, 11 | tcphsca 25269 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝐺) |
13 | 12 | a1i 11 | . . 3 ⊢ (⊤ → (Scalar‘𝑊) = (Scalar‘𝐺)) |
14 | eqid 2734 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
15 | eqid 2734 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
16 | 2, 15 | tcphvsca 25270 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝐺) |
17 | 16 | a1i 11 | . . . 4 ⊢ (⊤ → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝐺)) |
18 | 17 | oveqdr 7473 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥( ·𝑠 ‘𝑊)𝑦) = (𝑥( ·𝑠 ‘𝐺)𝑦)) |
19 | eqid 2734 | . . . . . 6 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
20 | 2, 19 | tcphip 25271 | . . . . 5 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝐺) |
21 | 20 | a1i 11 | . . . 4 ⊢ (⊤ → (·𝑖‘𝑊) = (·𝑖‘𝐺)) |
22 | 21 | oveqdr 7473 | . . 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 21691 | . 2 ⊢ (⊤ → (𝑊 ∈ PreHil ↔ 𝐺 ∈ PreHil)) |
24 | 23 | mptru 1544 | 1 ⊢ (𝑊 ∈ PreHil ↔ 𝐺 ∈ PreHil) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 ⊤wtru 1538 ∈ wcel 2103 ‘cfv 6572 Basecbs 17253 +gcplusg 17306 Scalarcsca 17309 ·𝑠 cvsca 17310 ·𝑖cip 17311 PreHilcphl 21660 toℂPreHilctcph 25213 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 ax-pre-sup 11258 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5021 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-er 8759 df-map 8882 df-en 9000 df-dom 9001 df-sdom 9002 df-sup 9507 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-div 11944 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-rp 13054 df-seq 14049 df-exp 14109 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ds 17328 df-0g 17496 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-mhm 18813 df-grp 18971 df-ghm 19248 df-mgp 20157 df-ur 20204 df-ring 20257 df-lmod 20877 df-lmhm 21039 df-lvec 21120 df-sra 21190 df-rgmod 21191 df-phl 21662 df-tng 24611 df-tcph 25215 |
This theorem is referenced by: tcphcph 25283 |
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