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Mirrors > Home > MPE Home > Th. List > hlmulid | Structured version Visualization version GIF version |
Description: Hilbert space scalar multiplication by one. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlmulf.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
hlmulf.4 | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
Ref | Expression |
---|---|
hlmulid | ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlnv 28652 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec) | |
2 | hlmulf.1 | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | hlmulf.4 | . . 3 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) | |
4 | 2, 3 | nvsid 28388 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴) |
5 | 1, 4 | sylan 582 | 1 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6341 (class class class)co 7142 1c1 10524 NrmCVeccnv 28345 BaseSetcba 28347 ·𝑠OLD cns 28348 CHilOLDchlo 28646 |
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 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-ov 7145 df-oprab 7146 df-1st 7675 df-2nd 7676 df-vc 28320 df-nv 28353 df-va 28356 df-ba 28357 df-sm 28358 df-0v 28359 df-nmcv 28361 df-cbn 28624 df-hlo 28647 |
This theorem is referenced by: axhvmulid-zf 28749 |
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