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Theorem braadd 32034

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 31172	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝐵 +_ℎ 𝐶) ·_ih 𝐴) = ((𝐵 ·_ih 𝐴) + (𝐶 ·_ih 𝐴)))
2	1	3comr 1126	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐵 +_ℎ 𝐶) ·_ih 𝐴) = ((𝐵 ·_ih 𝐴) + (𝐶 ·_ih 𝐴)))
3		hvaddcl 31101	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +_ℎ 𝐶) ∈ ℋ)
4		braval 32033	. . . 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 1115	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘(𝐵 +_ℎ 𝐶)) = ((𝐵 +_ℎ 𝐶) ·_ih 𝐴))
7		braval 32033	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·_ih 𝐴))
8	7	3adant3 1133	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐵) = (𝐵 ·_ih 𝐴))
9		braval 32033	. . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐶) = (𝐶 ·_ih 𝐴))
10	9	3adant2 1132	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐴)‘𝐶) = (𝐶 ·_ih 𝐴))
11	8, 10	oveq12d 7379	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (((bra‘𝐴)‘𝐵) + ((bra‘𝐴)‘𝐶)) = ((𝐵 ·_ih 𝐴) + (𝐶 ·_ih 𝐴)))
12	2, 6, 11	3eqtr4d 2782	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 1087 = wceq 1542 ∈ wcel 2114 ‘cfv 6493 (class class class)co 7361 + caddc 11035 ℋchba 31008 +_ℎ cva 31009 ·_ih csp 31011 bracbr 31045

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pr 5371 ax-hilex 31088 ax-hfvadd 31089 ax-his2 31172

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7364 df-bra 31939

This theorem is referenced by: bralnfn 32037

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