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Theorem subval 7367
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 7366 . . . 4 ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)
2 riotacl 5513 . . . 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 264 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ ℂ)
5 eqeq2 2091 . . . 4 (𝑦 = 𝐴 → ((𝑧 + 𝑥) = 𝑦 ↔ (𝑧 + 𝑥) = 𝐴))
65riotabidv 5501 . . 3 (𝑦 = 𝐴 → (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝑦) = (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝐴))
7 oveq1 5550 . . . . 5 (𝑧 = 𝐵 → (𝑧 + 𝑥) = (𝐵 + 𝑥))
87eqeq1d 2090 . . . 4 (𝑧 = 𝐵 → ((𝑧 + 𝑥) = 𝐴 ↔ (𝐵 + 𝑥) = 𝐴))
98riotabidv 5501 . . 3 (𝑧 = 𝐵 → (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝐴) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
10 df-sub 7348 . . 3 − = (𝑦 ∈ ℂ, 𝑧 ∈ ℂ ↦ (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝑦))
116, 9, 10ovmpt2g 5666 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ ℂ) → (𝐴𝐵) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
124, 11mpd3an3 1270 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐵) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  ∃!wreu 2351  crio 5498  (class class class)co 5543  cc 7041   + caddc 7046  cmin 7346
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288  ax-resscn 7130  ax-1cn 7131  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-sub 7348
This theorem is referenced by:  subcl  7374  subf  7377  subadd  7378
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