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Mirrors > Home > HSE Home > Th. List > his7 | Structured version Visualization version GIF version |
Description: Distributive law for inner product. Lemma 3.1(S7) of [Beran] p. 95. (Contributed by NM, 31-Jul-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
his7 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ih (𝐵 +ℎ 𝐶)) = ((𝐴 ·ih 𝐵) + (𝐴 ·ih 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-his2 28862 | . . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 +ℎ 𝐶) ·ih 𝐴) = ((𝐵 ·ih 𝐴) + (𝐶 ·ih 𝐴))) | |
2 | 1 | fveq2d 6676 | . . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 +ℎ 𝐶) ·ih 𝐴)) = (∗‘((𝐵 ·ih 𝐴) + (𝐶 ·ih 𝐴)))) |
3 | hicl 28859 | . . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐵 ·ih 𝐴) ∈ ℂ) | |
4 | hicl 28859 | . . . . . 6 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐶 ·ih 𝐴) ∈ ℂ) | |
5 | cjadd 14502 | . . . . . 6 ⊢ (((𝐵 ·ih 𝐴) ∈ ℂ ∧ (𝐶 ·ih 𝐴) ∈ ℂ) → (∗‘((𝐵 ·ih 𝐴) + (𝐶 ·ih 𝐴))) = ((∗‘(𝐵 ·ih 𝐴)) + (∗‘(𝐶 ·ih 𝐴)))) | |
6 | 3, 4, 5 | syl2an 597 | . . . . 5 ⊢ (((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ)) → (∗‘((𝐵 ·ih 𝐴) + (𝐶 ·ih 𝐴))) = ((∗‘(𝐵 ·ih 𝐴)) + (∗‘(𝐶 ·ih 𝐴)))) |
7 | 6 | 3impdir 1347 | . . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 ·ih 𝐴) + (𝐶 ·ih 𝐴))) = ((∗‘(𝐵 ·ih 𝐴)) + (∗‘(𝐶 ·ih 𝐴)))) |
8 | 2, 7 | eqtrd 2858 | . . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 +ℎ 𝐶) ·ih 𝐴)) = ((∗‘(𝐵 ·ih 𝐴)) + (∗‘(𝐶 ·ih 𝐴)))) |
9 | 8 | 3comr 1121 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘((𝐵 +ℎ 𝐶) ·ih 𝐴)) = ((∗‘(𝐵 ·ih 𝐴)) + (∗‘(𝐶 ·ih 𝐴)))) |
10 | hvaddcl 28791 | . . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +ℎ 𝐶) ∈ ℋ) | |
11 | ax-his1 28861 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 +ℎ 𝐶) ∈ ℋ) → (𝐴 ·ih (𝐵 +ℎ 𝐶)) = (∗‘((𝐵 +ℎ 𝐶) ·ih 𝐴))) | |
12 | 10, 11 | sylan2 594 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → (𝐴 ·ih (𝐵 +ℎ 𝐶)) = (∗‘((𝐵 +ℎ 𝐶) ·ih 𝐴))) |
13 | 12 | 3impb 1111 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ih (𝐵 +ℎ 𝐶)) = (∗‘((𝐵 +ℎ 𝐶) ·ih 𝐴))) |
14 | ax-his1 28861 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴))) | |
15 | 14 | 3adant3 1128 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴))) |
16 | ax-his1 28861 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ih 𝐶) = (∗‘(𝐶 ·ih 𝐴))) | |
17 | 16 | 3adant2 1127 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ih 𝐶) = (∗‘(𝐶 ·ih 𝐴))) |
18 | 15, 17 | oveq12d 7176 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·ih 𝐵) + (𝐴 ·ih 𝐶)) = ((∗‘(𝐵 ·ih 𝐴)) + (∗‘(𝐶 ·ih 𝐴)))) |
19 | 9, 13, 18 | 3eqtr4d 2868 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ih (𝐵 +ℎ 𝐶)) = ((𝐴 ·ih 𝐵) + (𝐴 ·ih 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 + caddc 10542 ∗ccj 14457 ℋchba 28698 +ℎ cva 28699 ·ih csp 28701 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-hfvadd 28779 ax-hfi 28858 ax-his1 28861 ax-his2 28862 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-2 11703 df-cj 14460 df-re 14461 df-im 14462 |
This theorem is referenced by: normlem0 28888 normlem8 28896 pjadjii 29453 lnopunilem1 29789 hmops 29799 cnlnadjlem6 29851 adjlnop 29865 adjadd 29872 hstoh 30011 |
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