Theorem addrval 45296

Description: Value of the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.)

Assertion

Ref	Expression
addrval	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) + (𝐵‘𝑣))))

Distinct variable groups:   𝑣,𝐴   𝑣,𝐵

Allowed substitution hints:   𝐶(𝑣)   𝐷(𝑣)

Proof of Theorem addrval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3474	. 2 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V)
2		elex 3474	. 2 ⊢ (𝐵 ∈ 𝐷 → 𝐵 ∈ V)
3		fveq1 6881	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥‘𝑣) = (𝐴‘𝑣))
4		fveq1 6881	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦‘𝑣) = (𝐵‘𝑣))
5	3, 4	oveqan12d 7436	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥‘𝑣) + (𝑦‘𝑣)) = ((𝐴‘𝑣) + (𝐵‘𝑣)))
6	5	mpteq2dv 5203	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) + (𝑦‘𝑣))) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) + (𝐵‘𝑣))))
7		df-addr 45293	. . 3 ⊢ +𝑟 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑣 ∈ ℝ ↦ ((𝑥‘𝑣) + (𝑦‘𝑣))))
8		reex 11219	. . . 4 ⊢ ℝ ∈ V
9	8	mptex 7226	. . 3 ⊢ (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) + (𝐵‘𝑣))) ∈ V
10	6, 7, 9	ovmpoa 7572	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴+𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) + (𝐵‘𝑣))))
11	1, 2, 10	syl2an 608	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴+𝑟𝐵) = (𝑣 ∈ ℝ ↦ ((𝐴‘𝑣) + (𝐵‘𝑣))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ↦ cmpt 5190  ‘cfv 6537  (class class class)co 7417  ℝcr 11127   + caddc 11131  +_𝑟cplusr 45287

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-cnex 11184  ax-resscn 11185

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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-ov 7420  df-oprab 7421  df-mpo 7422  df-addr 45293

This theorem is used by:  addrfv  45299  addrfn  45302