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Mirrors > Home > HSE Home > Th. List > hi02 | Structured version Visualization version GIF version |
Description: Inner product with the 0 vector. (Contributed by NM, 13-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hi02 | ⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 0ℎ) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hv0cl 29361 | . . 3 ⊢ 0ℎ ∈ ℋ | |
2 | ax-his1 29440 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝐴 ·ih 0ℎ) = (∗‘(0ℎ ·ih 𝐴))) | |
3 | 1, 2 | mpan2 688 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 0ℎ) = (∗‘(0ℎ ·ih 𝐴))) |
4 | hi01 29454 | . . . 4 ⊢ (𝐴 ∈ ℋ → (0ℎ ·ih 𝐴) = 0) | |
5 | 4 | fveq2d 6775 | . . 3 ⊢ (𝐴 ∈ ℋ → (∗‘(0ℎ ·ih 𝐴)) = (∗‘0)) |
6 | cj0 14867 | . . 3 ⊢ (∗‘0) = 0 | |
7 | 5, 6 | eqtrdi 2796 | . 2 ⊢ (𝐴 ∈ ℋ → (∗‘(0ℎ ·ih 𝐴)) = 0) |
8 | 3, 7 | eqtrd 2780 | 1 ⊢ (𝐴 ∈ ℋ → (𝐴 ·ih 0ℎ) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 ‘cfv 6432 (class class class)co 7271 0cc0 10872 ∗ccj 14805 ℋchba 29277 ·ih csp 29280 0ℎc0v 29282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-hv0cl 29361 ax-hvmul0 29368 ax-hfi 29437 ax-his1 29440 ax-his3 29442 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-po 5504 df-so 5505 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-2 12036 df-cj 14808 df-re 14809 df-im 14810 |
This theorem is referenced by: hial02 29461 choc0 29684 bra0 30308 0hmop 30341 adj0 30352 riesz3i 30420 |
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