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Theorem lnfnaddmuli 32338

Description: Sum/product property of a linear Hilbert space functional. (Contributed by NM, 13-Feb-2006.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
lnfnl.1	⊢ 𝑇 ∈ LinFn

**Assertion**
Ref	Expression
lnfnaddmuli	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 +_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) + (𝐴 · (𝑇‘𝐶))))

**Proof of Theorem lnfnaddmuli**
Step	Hyp	Ref	Expression
1		hvmulcl 31306	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
2		lnfnl.1	. . . . . 6 ⊢ 𝑇 ∈ LinFn
3	2	lnfnaddi 32336	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ·_ℎ 𝐶) ∈ ℋ) → (𝑇‘(𝐵 +_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) + (𝑇‘(𝐴 ·_ℎ 𝐶))))
4	1, 3	sylan2 604	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ)) → (𝑇‘(𝐵 +_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) + (𝑇‘(𝐴 ·_ℎ 𝐶))))
5	4	3impb 1130	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 +_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) + (𝑇‘(𝐴 ·_ℎ 𝐶))))
6	5	3com12 1139	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 +_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) + (𝑇‘(𝐴 ·_ℎ 𝐶))))
7	2	lnfnmuli 32337	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐴 ·_ℎ 𝐶)) = (𝐴 · (𝑇‘𝐶)))
8	7	3adant2 1147	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐴 ·_ℎ 𝐶)) = (𝐴 · (𝑇‘𝐶)))
9	8	oveq2d 7427	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝑇‘𝐵) + (𝑇‘(𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) + (𝐴 · (𝑇‘𝐶))))
10	6, 9	eqtrd 2804	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 +_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) + (𝐴 · (𝑇‘𝐶))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ‘cfv 6537 (class class class)co 7411 ℂcc 11098 + caddc 11103 · cmul 11105 ℋchba 31212 +_ℎ cva 31213 ·_ℎ csm 31214 LinFnclf 31247

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5259 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-hilex 31292 ax-hv0cl 31296 ax-hvaddid 31297 ax-hfvmul 31298 ax-hvmulid 31299

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-reu 3376 df-rab 3423 df-v 3463 df-sbc 3752 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5557 df-po 5570 df-so 5571 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-er 8694 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11245 df-mnf 11246 df-ltxr 11248 df-sub 11443 df-lnfn 32141

This theorem is referenced by: lnfnsubi 32339

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