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Mirrors > Home > HSE Home > Th. List > braadd | Structured version Visualization version GIF version |
Description: Linearity property of bra for addition. (Contributed by NM, 23-May-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
braadd | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 +ℎ 𝐶)) = (((bra‘𝐴)‘𝐵) + ((bra‘𝐴)‘𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-his2 28860 | . . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 +ℎ 𝐶) ·ih 𝐴) = ((𝐵 ·ih 𝐴) + (𝐶 ·ih 𝐴))) | |
2 | 1 | 3comr 1121 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 +ℎ 𝐶) ·ih 𝐴) = ((𝐵 ·ih 𝐴) + (𝐶 ·ih 𝐴))) |
3 | hvaddcl 28789 | . . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +ℎ 𝐶) ∈ ℋ) | |
4 | braval 29721 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 +ℎ 𝐶) ∈ ℋ) → ((bra‘𝐴)‘(𝐵 +ℎ 𝐶)) = ((𝐵 +ℎ 𝐶) ·ih 𝐴)) | |
5 | 3, 4 | sylan2 594 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → ((bra‘𝐴)‘(𝐵 +ℎ 𝐶)) = ((𝐵 +ℎ 𝐶) ·ih 𝐴)) |
6 | 5 | 3impb 1111 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 +ℎ 𝐶)) = ((𝐵 +ℎ 𝐶) ·ih 𝐴)) |
7 | braval 29721 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·ih 𝐴)) | |
8 | 7 | 3adant3 1128 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·ih 𝐴)) |
9 | braval 29721 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐶) = (𝐶 ·ih 𝐴)) | |
10 | 9 | 3adant2 1127 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐶) = (𝐶 ·ih 𝐴)) |
11 | 8, 10 | oveq12d 7174 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (((bra‘𝐴)‘𝐵) + ((bra‘𝐴)‘𝐶)) = ((𝐵 ·ih 𝐴) + (𝐶 ·ih 𝐴))) |
12 | 2, 6, 11 | 3eqtr4d 2866 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 +ℎ 𝐶)) = (((bra‘𝐴)‘𝐵) + ((bra‘𝐴)‘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 + caddc 10540 ℋchba 28696 +ℎ cva 28697 ·ih csp 28699 bracbr 28733 |
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-rep 5190 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-hilex 28776 ax-hfvadd 28777 ax-his2 28860 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3143 df-rex 3144 df-reu 3145 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-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-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-ov 7159 df-bra 29627 |
This theorem is referenced by: bralnfn 29725 |
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