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Theorem subval 7871
Description: Value of subtraction, which is the (unique) element 𝑥 such that 𝐵 + 𝑥 = 𝐴. (Contributed by NM, 4-Aug-2007.) (Revised by Mario Carneiro, 2-Nov-2013.)
Assertion
Ref Expression
subval ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐵) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem subval
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 negeu 7870 . . . 4 ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)
2 riotacl 5696 . . . 4 (∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴 → (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ ℂ)
31, 2syl 14 . . 3 ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ ℂ)
43ancoms 266 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ ℂ)
5 eqeq2 2122 . . . 4 (𝑦 = 𝐴 → ((𝑧 + 𝑥) = 𝑦 ↔ (𝑧 + 𝑥) = 𝐴))
65riotabidv 5684 . . 3 (𝑦 = 𝐴 → (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝑦) = (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝐴))
7 oveq1 5733 . . . . 5 (𝑧 = 𝐵 → (𝑧 + 𝑥) = (𝐵 + 𝑥))
87eqeq1d 2121 . . . 4 (𝑧 = 𝐵 → ((𝑧 + 𝑥) = 𝐴 ↔ (𝐵 + 𝑥) = 𝐴))
98riotabidv 5684 . . 3 (𝑧 = 𝐵 → (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝐴) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
10 df-sub 7852 . . 3 − = (𝑦 ∈ ℂ, 𝑧 ∈ ℂ ↦ (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝑦))
116, 9, 10ovmpog 5857 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ ℂ) → (𝐴𝐵) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
124, 11mpd3an3 1297 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐵) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1312  wcel 1461  ∃!wreu 2390  crio 5681  (class class class)co 5726  cc 7539   + caddc 7544  cmin 7850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-setind 4410  ax-resscn 7631  ax-1cn 7632  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-addcom 7639  ax-addass 7641  ax-distr 7643  ax-i2m1 7644  ax-0id 7647  ax-rnegex 7648  ax-cnre 7650
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-sub 7852
This theorem is referenced by:  subcl  7878  subf  7881  subadd  7882
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