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Theorem subrfv 38494
 Description: Vector subtraction at a value. (Contributed by Andrew Salmon, 27-Jan-2012.)
Assertion
Ref Expression
subrfv ((𝐴𝐸𝐵𝐷𝐶 ∈ ℝ) → ((𝐴-𝑟𝐵)‘𝐶) = ((𝐴𝐶) − (𝐵𝐶)))

Proof of Theorem subrfv
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 subrval 38491 . . . 4 ((𝐴𝐸𝐵𝐷) → (𝐴-𝑟𝐵) = (𝑥 ∈ ℝ ↦ ((𝐴𝑥) − (𝐵𝑥))))
21fveq1d 6180 . . 3 ((𝐴𝐸𝐵𝐷) → ((𝐴-𝑟𝐵)‘𝐶) = ((𝑥 ∈ ℝ ↦ ((𝐴𝑥) − (𝐵𝑥)))‘𝐶))
3 fveq2 6178 . . . . 5 (𝑥 = 𝐶 → (𝐴𝑥) = (𝐴𝐶))
4 fveq2 6178 . . . . 5 (𝑥 = 𝐶 → (𝐵𝑥) = (𝐵𝐶))
53, 4oveq12d 6653 . . . 4 (𝑥 = 𝐶 → ((𝐴𝑥) − (𝐵𝑥)) = ((𝐴𝐶) − (𝐵𝐶)))
6 eqid 2620 . . . 4 (𝑥 ∈ ℝ ↦ ((𝐴𝑥) − (𝐵𝑥))) = (𝑥 ∈ ℝ ↦ ((𝐴𝑥) − (𝐵𝑥)))
7 ovex 6663 . . . 4 ((𝐴𝐶) − (𝐵𝐶)) ∈ V
85, 6, 7fvmpt 6269 . . 3 (𝐶 ∈ ℝ → ((𝑥 ∈ ℝ ↦ ((𝐴𝑥) − (𝐵𝑥)))‘𝐶) = ((𝐴𝐶) − (𝐵𝐶)))
92, 8sylan9eq 2674 . 2 (((𝐴𝐸𝐵𝐷) ∧ 𝐶 ∈ ℝ) → ((𝐴-𝑟𝐵)‘𝐶) = ((𝐴𝐶) − (𝐵𝐶)))
1093impa 1257 1 ((𝐴𝐸𝐵𝐷𝐶 ∈ ℝ) → ((𝐴-𝑟𝐵)‘𝐶) = ((𝐴𝐶) − (𝐵𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1481   ∈ wcel 1988   ↦ cmpt 4720  ‘cfv 5876  (class class class)co 6635  ℝcr 9920   − cmin 10251  -𝑟cminusr 38482 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-cnex 9977  ax-resscn 9978 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-subr 38488 This theorem is referenced by: (None)
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