Theorem hfsval 30105

Description: Value of the sum of two Hilbert space functionals. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
hfsval	⊢ ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → ((𝑆 +fn 𝑇)‘𝐴) = ((𝑆‘𝐴) + (𝑇‘𝐴)))

Proof of Theorem hfsval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hfsmval 30100	. . . 4 ⊢ ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))))
2	1	fveq1d 6776	. . 3 ⊢ ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) → ((𝑆 +fn 𝑇)‘𝐴) = ((𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥)))‘𝐴))
3		fveq2 6774	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴))
4		fveq2 6774	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴))
5	3, 4	oveq12d 7293	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) + (𝑇‘𝑥)) = ((𝑆‘𝐴) + (𝑇‘𝐴)))
6		eqid 2738	. . . 4 ⊢ (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥)))
7		ovex 7308	. . . 4 ⊢ ((𝑆‘𝐴) + (𝑇‘𝐴)) ∈ V
8	5, 6, 7	fvmpt 6875	. . 3 ⊢ (𝐴 ∈ ℋ → ((𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥)))‘𝐴) = ((𝑆‘𝐴) + (𝑇‘𝐴)))
9	2, 8	sylan9eq 2798	. 2 ⊢ (((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) ∧ 𝐴 ∈ ℋ) → ((𝑆 +fn 𝑇)‘𝐴) = ((𝑆‘𝐴) + (𝑇‘𝐴)))
10	9	3impa 1109	1 ⊢ ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → ((𝑆 +fn 𝑇)‘𝐴) = ((𝑆‘𝐴) + (𝑇‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ↦ cmpt 5157  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  ℂcc 10869   + caddc 10874   ℋchba 29281   +_fn chfs 29303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-hilex 29361

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-hfsum 30095

This theorem is referenced by: (None)