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Mirrors > Home > MPE Home > Th. List > nvvop | Structured version Visualization version GIF version |
Description: The vector space component of a normed complex vector space is an ordered pair of the underlying group and a scalar product. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvvop.1 | ⊢ 𝑊 = (1st ‘𝑈) |
nvvop.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
nvvop.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
Ref | Expression |
---|---|
nvvop | ⊢ (𝑈 ∈ NrmCVec → 𝑊 = 〈𝐺, 𝑆〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vcrel 27415 | . . 3 ⊢ Rel CVecOLD | |
2 | nvss 27448 | . . . . 5 ⊢ NrmCVec ⊆ (CVecOLD × V) | |
3 | nvvop.1 | . . . . . . . 8 ⊢ 𝑊 = (1st ‘𝑈) | |
4 | eqid 2622 | . . . . . . . 8 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
5 | 3, 4 | nvop2 27463 | . . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = 〈𝑊, (normCV‘𝑈)〉) |
6 | 5 | eleq1d 2686 | . . . . . 6 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ↔ 〈𝑊, (normCV‘𝑈)〉 ∈ NrmCVec)) |
7 | 6 | ibi 256 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → 〈𝑊, (normCV‘𝑈)〉 ∈ NrmCVec) |
8 | 2, 7 | sseldi 3601 | . . . 4 ⊢ (𝑈 ∈ NrmCVec → 〈𝑊, (normCV‘𝑈)〉 ∈ (CVecOLD × V)) |
9 | opelxp1 5150 | . . . 4 ⊢ (〈𝑊, (normCV‘𝑈)〉 ∈ (CVecOLD × V) → 𝑊 ∈ CVecOLD) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑊 ∈ CVecOLD) |
11 | 1st2nd 7214 | . . 3 ⊢ ((Rel CVecOLD ∧ 𝑊 ∈ CVecOLD) → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) | |
12 | 1, 10, 11 | sylancr 695 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉) |
13 | nvvop.2 | . . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
14 | 13 | vafval 27458 | . . . 4 ⊢ 𝐺 = (1st ‘(1st ‘𝑈)) |
15 | 3 | fveq2i 6194 | . . . 4 ⊢ (1st ‘𝑊) = (1st ‘(1st ‘𝑈)) |
16 | 14, 15 | eqtr4i 2647 | . . 3 ⊢ 𝐺 = (1st ‘𝑊) |
17 | nvvop.4 | . . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
18 | 17 | smfval 27460 | . . . 4 ⊢ 𝑆 = (2nd ‘(1st ‘𝑈)) |
19 | 3 | fveq2i 6194 | . . . 4 ⊢ (2nd ‘𝑊) = (2nd ‘(1st ‘𝑈)) |
20 | 18, 19 | eqtr4i 2647 | . . 3 ⊢ 𝑆 = (2nd ‘𝑊) |
21 | 16, 20 | opeq12i 4407 | . 2 ⊢ 〈𝐺, 𝑆〉 = 〈(1st ‘𝑊), (2nd ‘𝑊)〉 |
22 | 12, 21 | syl6eqr 2674 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑊 = 〈𝐺, 𝑆〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 〈cop 4183 × cxp 5112 Rel wrel 5119 ‘cfv 5888 1st c1st 7166 2nd c2nd 7167 CVecOLDcvc 27413 NrmCVeccnv 27439 +𝑣 cpv 27440 ·𝑠OLD cns 27442 normCVcnmcv 27445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fo 5894 df-fv 5896 df-oprab 6654 df-1st 7168 df-2nd 7169 df-vc 27414 df-nv 27447 df-va 27450 df-sm 27452 df-nmcv 27455 |
This theorem is referenced by: nvi 27469 nvvc 27470 nvop 27531 |
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