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Mirrors > Home > MPE Home > Th. List > hlmul0 | Structured version Visualization version GIF version |
Description: Hilbert space scalar multiplication by zero. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlmul0.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
hlmul0.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
hlmul0.5 | ⊢ 𝑍 = (0vec‘𝑈) |
Ref | Expression |
---|---|
hlmul0 | ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlnv 30914 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec) | |
2 | hlmul0.1 | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | hlmul0.4 | . . 3 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
4 | hlmul0.5 | . . 3 ⊢ 𝑍 = (0vec‘𝑈) | |
5 | 2, 3, 4 | nv0 30660 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍) |
6 | 1, 5 | sylan 579 | 1 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋) → (0𝑆𝐴) = 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2103 ‘cfv 6572 (class class class)co 7445 0cc0 11180 NrmCVeccnv 30607 BaseSetcba 30609 ·𝑠OLD cns 30610 0veccn0v 30611 CHilOLDchlo 30908 |
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-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 |
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-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-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-id 5597 df-po 5611 df-so 5612 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-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-1st 8026 df-2nd 8027 df-er 8759 df-en 9000 df-dom 9001 df-sdom 9002 df-pnf 11322 df-mnf 11323 df-ltxr 11325 df-grpo 30516 df-gid 30517 df-ginv 30518 df-ablo 30568 df-vc 30582 df-nv 30615 df-va 30618 df-ba 30619 df-sm 30620 df-0v 30621 df-nmcv 30623 df-cbn 30886 df-hlo 30909 |
This theorem is referenced by: axhvmul0-zf 31015 |
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