Theorem hfsval 32354

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 32349	. . . 4 ⊢ ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝑆 +fn 𝑇) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))))
2	1	fveq1d 6892	. . 3 ⊢ ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) → ((𝑆 +fn 𝑇)‘𝐴) = ((𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥)))‘𝐴))
3		fveq2 6890	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑆‘𝑥) = (𝑆‘𝐴))
4		fveq2 6890	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑇‘𝑥) = (𝑇‘𝐴))
5	3, 4	oveq12d 7442	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑆‘𝑥) + (𝑇‘𝑥)) = ((𝑆‘𝐴) + (𝑇‘𝐴)))
6		eqid 2760	. . . 4 ⊢ (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥))) = (𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥)))
7		ovex 7457	. . . 4 ⊢ ((𝑆‘𝐴) + (𝑇‘𝐴)) ∈ V
8	5, 6, 7	fvmpt 6998	. . 3 ⊢ (𝐴 ∈ ℋ → ((𝑥 ∈ ℋ ↦ ((𝑆‘𝑥) + (𝑇‘𝑥)))‘𝐴) = ((𝑆‘𝐴) + (𝑇‘𝐴)))
9	2, 8	sylan9eq 2815	. 2 ⊢ (((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ) ∧ 𝐴 ∈ ℋ) → ((𝑆 +fn 𝑇)‘𝐴) = ((𝑆‘𝐴) + (𝑇‘𝐴)))
10	9	3impa 1127	1 ⊢ ((𝑆: ℋ⟶ℂ ∧ 𝑇: ℋ⟶ℂ ∧ 𝐴 ∈ ℋ) → ((𝑆 +fn 𝑇)‘𝐴) = ((𝑆‘𝐴) + (𝑇‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ↦ cmpt 5185  ⟶wf 6539  ‘cfv 6543  (class class class)co 7424  ℂcc 11205   + caddc 11210   ℋchba 31530   +_fn chfs 31552

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-hilex 31610

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-map 8856  df-hfsum 32344

This theorem is used by: (None)