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Theorem braadd 31969

Description: Linearity property of bra for addition. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
braadd	⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 +_ℎ 𝐶)) = (((bra‘𝐴)‘𝐵) + ((bra‘𝐴)‘𝐶)))

**Proof of Theorem braadd**
Step	Hyp	Ref	Expression
1		ax-his2 31107	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 +_ℎ 𝐶) ·_ih 𝐴) = ((𝐵 ·_ih 𝐴) + (𝐶 ·_ih 𝐴)))
2	1	3comr 1125	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 +_ℎ 𝐶) ·_ih 𝐴) = ((𝐵 ·_ih 𝐴) + (𝐶 ·_ih 𝐴)))
3		hvaddcl 31036	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +_ℎ 𝐶) ∈ ℋ)
4		braval 31968	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 +_ℎ 𝐶) ∈ ℋ) → ((bra‘𝐴)‘(𝐵 +_ℎ 𝐶)) = ((𝐵 +_ℎ 𝐶) ·_ih 𝐴))
5	3, 4	sylan2 593	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)) → ((bra‘𝐴)‘(𝐵 +_ℎ 𝐶)) = ((𝐵 +_ℎ 𝐶) ·_ih 𝐴))
6	5	3impb 1114	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 +_ℎ 𝐶)) = ((𝐵 +_ℎ 𝐶) ·_ih 𝐴))
7		braval 31968	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·_ih 𝐴))
8	7	3adant3 1132	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·_ih 𝐴))
9		braval 31968	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐶) = (𝐶 ·_ih 𝐴))
10	9	3adant2 1131	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐶) = (𝐶 ·_ih 𝐴))
11	8, 10	oveq12d 7374	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (((bra‘𝐴)‘𝐵) + ((bra‘𝐴)‘𝐶)) = ((𝐵 ·_ih 𝐴) + (𝐶 ·_ih 𝐴)))
12	2, 6, 11	3eqtr4d 2779	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 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ‘cfv 6490 (class class class)co 7356 + caddc 11027 ℋchba 30943 +_ℎ cva 30944 ·_ih csp 30946 bracbr 30980

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pr 5375 ax-hilex 31023 ax-hfvadd 31024 ax-his2 31107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7359 df-bra 31874

This theorem is referenced by: bralnfn 31972

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