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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 10805 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴 · 0) = 0) | |
2 | 1 | oveq1d 7157 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0) ·ℎ 0ℎ) = (0 ·ℎ 0ℎ)) |
3 | ax-hv0cl 28764 | . . . . 5 ⊢ 0ℎ ∈ ℋ | |
4 | ax-hvmul0 28771 | . . . . 5 ⊢ (0ℎ ∈ ℋ → (0 ·ℎ 0ℎ) = 0ℎ) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ (0 ·ℎ 0ℎ) = 0ℎ |
6 | 2, 5 | syl6eq 2872 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0) ·ℎ 0ℎ) = 0ℎ) |
7 | 0cn 10619 | . . . 4 ⊢ 0 ∈ ℂ | |
8 | ax-hvmulass 28768 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 0ℎ ∈ ℋ) → ((𝐴 · 0) ·ℎ 0ℎ) = (𝐴 ·ℎ (0 ·ℎ 0ℎ))) | |
9 | 7, 3, 8 | mp3an23 1449 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴 · 0) ·ℎ 0ℎ) = (𝐴 ·ℎ (0 ·ℎ 0ℎ))) |
10 | 6, 9 | eqtr3d 2858 | . 2 ⊢ (𝐴 ∈ ℂ → 0ℎ = (𝐴 ·ℎ (0 ·ℎ 0ℎ))) |
11 | 5 | oveq2i 7153 | . 2 ⊢ (𝐴 ·ℎ (0 ·ℎ 0ℎ)) = (𝐴 ·ℎ 0ℎ) |
12 | 10, 11 | syl6req 2873 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 ·ℎ 0ℎ) = 0ℎ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 (class class class)co 7142 ℂcc 10521 0cc0 10523 · cmul 10528 ℋchba 28680 ·ℎ csm 28682 0ℎc0v 28685 |
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-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-hv0cl 28764 ax-hvmulass 28768 ax-hvmul0 28771 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-nel 3124 df-ral 3143 df-rex 3144 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-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-po 5460 df-so 5461 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-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-pnf 10663 df-mnf 10664 df-ltxr 10666 |
This theorem is referenced by: hvmul0or 28786 hvsub0 28837 hsn0elch 29009 pjssmii 29442 |
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