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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 2822 | . . 3 ⊢ (⊤ → (Base‘𝑊) = (Base‘𝑊)) | |
2 | tcphval.n | . . . . 5 ⊢ 𝐺 = (toℂPreHil‘𝑊) | |
3 | eqid 2821 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | 2, 3 | tcphbas 23822 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝐺) |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → (Base‘𝑊) = (Base‘𝐺)) |
6 | eqid 2821 | . . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
7 | 2, 6 | tchplusg 23823 | . . . . 5 ⊢ (+g‘𝑊) = (+g‘𝐺) |
8 | 7 | a1i 11 | . . . 4 ⊢ (⊤ → (+g‘𝑊) = (+g‘𝐺)) |
9 | 8 | oveqdr 7184 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐺)𝑦)) |
10 | eqidd 2822 | . . 3 ⊢ (⊤ → (Scalar‘𝑊) = (Scalar‘𝑊)) | |
11 | eqid 2821 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
12 | 2, 11 | tcphsca 23826 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝐺) |
13 | 12 | a1i 11 | . . 3 ⊢ (⊤ → (Scalar‘𝑊) = (Scalar‘𝐺)) |
14 | eqid 2821 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
15 | eqid 2821 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
16 | 2, 15 | tcphvsca 23827 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝐺) |
17 | 16 | a1i 11 | . . . 4 ⊢ (⊤ → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝐺)) |
18 | 17 | oveqdr 7184 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥( ·𝑠 ‘𝑊)𝑦) = (𝑥( ·𝑠 ‘𝐺)𝑦)) |
19 | eqid 2821 | . . . . . 6 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
20 | 2, 19 | tcphip 23828 | . . . . 5 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝐺) |
21 | 20 | a1i 11 | . . . 4 ⊢ (⊤ → (·𝑖‘𝑊) = (·𝑖‘𝐺)) |
22 | 21 | oveqdr 7184 | . . 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 20799 | . 2 ⊢ (⊤ → (𝑊 ∈ PreHil ↔ 𝐺 ∈ PreHil)) |
24 | 23 | mptru 1544 | 1 ⊢ (𝑊 ∈ PreHil ↔ 𝐺 ∈ PreHil) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ‘cfv 6355 Basecbs 16483 +gcplusg 16565 Scalarcsca 16568 ·𝑠 cvsca 16569 ·𝑖cip 16570 PreHilcphl 20768 toℂPreHilctcph 23771 |
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-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 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-iun 4921 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-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 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-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ds 16587 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 df-grp 18106 df-ghm 18356 df-mgp 19240 df-ur 19252 df-ring 19299 df-lmod 19636 df-lmhm 19794 df-lvec 19875 df-sra 19944 df-rgmod 19945 df-phl 20770 df-tng 23194 df-tcph 23773 |
This theorem is referenced by: tcphcph 23840 |
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