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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 28860 | . . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 +ℎ 𝐶) ·ih 𝐴) = ((𝐵 ·ih 𝐴) + (𝐶 ·ih 𝐴))) | |
2 | 1 | fveq2d 6674 | . . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (∗‘((𝐵 +ℎ 𝐶) ·ih 𝐴)) = (∗‘((𝐵 ·ih 𝐴) + (𝐶 ·ih 𝐴)))) |
3 | hicl 28857 | . . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐵 ·ih 𝐴) ∈ ℂ) | |
4 | hicl 28857 | . . . . . 6 ⊢ ((𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → (𝐶 ·ih 𝐴) ∈ ℂ) | |
5 | cjadd 14500 | . . . . . 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 2856 | . . 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 28789 | . . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +ℎ 𝐶) ∈ ℋ) | |
11 | ax-his1 28859 | . . . 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 28859 | . . . 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 28859 | . . . 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 7174 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ·ih 𝐵) + (𝐴 ·ih 𝐶)) = ((∗‘(𝐵 ·ih 𝐴)) + (∗‘(𝐶 ·ih 𝐴)))) |
19 | 9, 13, 18 | 3eqtr4d 2866 | 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 6355 (class class class)co 7156 ℂcc 10535 + caddc 10540 ∗ccj 14455 ℋchba 28696 +ℎ cva 28697 ·ih csp 28699 |
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 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-hfvadd 28777 ax-hfi 28856 ax-his1 28859 ax-his2 28860 |
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-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-2 11701 df-cj 14458 df-re 14459 df-im 14460 |
This theorem is referenced by: normlem0 28886 normlem8 28894 pjadjii 29451 lnopunilem1 29787 hmops 29797 cnlnadjlem6 29849 adjlnop 29863 adjadd 29870 hstoh 30009 |
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