![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > hvmul0 | Structured version Visualization version GIF version |
Description: Scalar multiplication with the zero vector. (Contributed by NM, 30-May-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvmul0 | ⊢ (𝐴 ∈ ℂ → (𝐴 ·ℎ 0ℎ) = 0ℎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mul01 11443 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0) | |
2 | 1 | oveq1d 7439 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0) ·ℎ 0ℎ) = (0 ·ℎ 0ℎ)) |
3 | ax-hv0cl 30936 | . . . . 5 ⊢ 0ℎ ∈ ℋ | |
4 | ax-hvmul0 30943 | . . . . 5 ⊢ (0ℎ ∈ ℋ → (0 ·ℎ 0ℎ) = 0ℎ) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ (0 ·ℎ 0ℎ) = 0ℎ |
6 | 2, 5 | eqtrdi 2782 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0) ·ℎ 0ℎ) = 0ℎ) |
7 | 0cn 11256 | . . . 4 ⊢ 0 ∈ ℂ | |
8 | ax-hvmulass 30940 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 0ℎ ∈ ℋ) → ((𝐴 · 0) ·ℎ 0ℎ) = (𝐴 ·ℎ (0 ·ℎ 0ℎ))) | |
9 | 7, 3, 8 | mp3an23 1450 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0) ·ℎ 0ℎ) = (𝐴 ·ℎ (0 ·ℎ 0ℎ))) |
10 | 6, 9 | eqtr3d 2768 | . 2 ⊢ (𝐴 ∈ ℂ → 0ℎ = (𝐴 ·ℎ (0 ·ℎ 0ℎ))) |
11 | 5 | oveq2i 7435 | . 2 ⊢ (𝐴 ·ℎ (0 ·ℎ 0ℎ)) = (𝐴 ·ℎ 0ℎ) |
12 | 10, 11 | eqtr2di 2783 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 ·ℎ 0ℎ) = 0ℎ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 (class class class)co 7424 ℂcc 11156 0cc0 11158 · cmul 11163 ℋchba 30852 ·ℎ csm 30854 0ℎc0v 30857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-hv0cl 30936 ax-hvmulass 30940 ax-hvmul0 30943 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-ltxr 11303 |
This theorem is referenced by: hvmul0or 30958 hvsub0 31009 hsn0elch 31181 pjssmii 31614 |
Copyright terms: Public domain | W3C validator |