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Theorem subval 7203
 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 7202 . . . 4 ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)
2 riotacl 5482 . . . 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 255 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ ℂ)
5 eqeq2 2049 . . . 4 (𝑦 = 𝐴 → ((𝑧 + 𝑥) = 𝑦 ↔ (𝑧 + 𝑥) = 𝐴))
65riotabidv 5470 . . 3 (𝑦 = 𝐴 → (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝑦) = (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝐴))
7 oveq1 5519 . . . . 5 (𝑧 = 𝐵 → (𝑧 + 𝑥) = (𝐵 + 𝑥))
87eqeq1d 2048 . . . 4 (𝑧 = 𝐵 → ((𝑧 + 𝑥) = 𝐴 ↔ (𝐵 + 𝑥) = 𝐴))
98riotabidv 5470 . . 3 (𝑧 = 𝐵 → (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝐴) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
10 df-sub 7184 . . 3 − = (𝑦 ∈ ℂ, 𝑧 ∈ ℂ ↦ (𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝑦))
116, 9, 10ovmpt2g 5635 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ ℂ) → (𝐴𝐵) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
124, 11mpd3an3 1233 1 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐵) = (𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ∃!wreu 2308  ℩crio 5467  (class class class)co 5512  ℂcc 6887   + caddc 6892   − cmin 7182 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262  ax-resscn 6976  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-sub 7184 This theorem is referenced by:  subcl  7210  subf  7213  subadd  7214
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