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Mirrors > Home > MPE Home > Th. List > tchphl | Structured version Visualization version GIF version |
Description: Augmentation of a 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 |
---|---|
tchval.n | ⊢ 𝐺 = (toℂHil‘𝑊) |
Ref | Expression |
---|---|
tchphl | ⊢ (𝑊 ∈ PreHil ↔ 𝐺 ∈ PreHil) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2611 | . . 3 ⊢ (⊤ → (Base‘𝑊) = (Base‘𝑊)) | |
2 | tchval.n | . . . . 5 ⊢ 𝐺 = (toℂHil‘𝑊) | |
3 | eqid 2610 | . . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
4 | 2, 3 | tchbas 22826 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝐺) |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → (Base‘𝑊) = (Base‘𝐺)) |
6 | eqid 2610 | . . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
7 | 2, 6 | tchplusg 22827 | . . . . 5 ⊢ (+g‘𝑊) = (+g‘𝐺) |
8 | 7 | a1i 11 | . . . 4 ⊢ (⊤ → (+g‘𝑊) = (+g‘𝐺)) |
9 | 8 | oveqdr 6573 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g‘𝑊)𝑦) = (𝑥(+g‘𝐺)𝑦)) |
10 | eqidd 2611 | . . 3 ⊢ (⊤ → (Scalar‘𝑊) = (Scalar‘𝑊)) | |
11 | eqid 2610 | . . . . 5 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
12 | 2, 11 | tchsca 22830 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝐺) |
13 | 12 | a1i 11 | . . 3 ⊢ (⊤ → (Scalar‘𝑊) = (Scalar‘𝐺)) |
14 | eqid 2610 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
15 | eqid 2610 | . . . . . 6 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
16 | 2, 15 | tchvsca 22831 | . . . . 5 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝐺) |
17 | 16 | a1i 11 | . . . 4 ⊢ (⊤ → ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝐺)) |
18 | 17 | oveqdr 6573 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥( ·𝑠 ‘𝑊)𝑦) = (𝑥( ·𝑠 ‘𝐺)𝑦)) |
19 | eqid 2610 | . . . . . 6 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
20 | 2, 19 | tchip 22832 | . . . . 5 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝐺) |
21 | 20 | a1i 11 | . . . 4 ⊢ (⊤ → (·𝑖‘𝑊) = (·𝑖‘𝐺)) |
22 | 21 | oveqdr 6573 | . . 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 19819 | . 2 ⊢ (⊤ → (𝑊 ∈ PreHil ↔ 𝐺 ∈ PreHil)) |
24 | 23 | trud 1484 | 1 ⊢ (𝑊 ∈ PreHil ↔ 𝐺 ∈ PreHil) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ‘cfv 5804 Basecbs 15695 +gcplusg 15768 Scalarcsca 15771 ·𝑠 cvsca 15772 ·𝑖cip 15773 PreHilcphl 19788 toℂHilctch 22775 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ds 15791 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-ghm 17481 df-mgp 18313 df-ur 18325 df-ring 18372 df-lmod 18688 df-lmhm 18843 df-lvec 18924 df-sra 18993 df-rgmod 18994 df-phl 19790 df-tng 22199 df-tch 22777 |
This theorem is referenced by: tchcph 22844 |
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