Theorem subval 8512

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) → (𝐴 − 𝐵) = (℩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.

Step	Hyp	Ref	Expression
1		negeu 8511	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)
2		riotacl 6048	. . . 4 ⊢ (∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴 → (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ ℂ)
3	1, 2	syl 14	. . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ ℂ)
4	3	ancoms 268	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ ℂ)
5		eqeq2 2248	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑧 + 𝑥) = 𝑦 ↔ (𝑧 + 𝑥) = 𝐴))
6	5	riotabidv 6034	. . 3 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝑦) = (℩𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝐴))
7		oveq1 6086	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑧 + 𝑥) = (𝐵 + 𝑥))
8	7	eqeq1d 2247	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝑧 + 𝑥) = 𝐴 ↔ (𝐵 + 𝑥) = 𝐴))
9	8	riotabidv 6034	. . 3 ⊢ (𝑧 = 𝐵 → (℩𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝐴) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
10		df-sub 8493	. . 3 ⊢ − = (𝑦 ∈ ℂ, 𝑧 ∈ ℂ ↦ (℩𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝑦))
11	6, 9, 10	ovmpog 6217	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ ℂ) → (𝐴 − 𝐵) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
12	4, 11	mpd3an3 1379	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∃!wreu 2530  ℩crio 6031  (class class class)co 6079  ℂcc 8171   + caddc 8176   − cmin 8491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-setind 4682  ax-resscn 8265  ax-1cn 8266  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-distr 8277  ax-i2m1 8278  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-iota 5335  df-fun 5377  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-sub 8493

This theorem is referenced by:  subcl  8519  subf  8522  subadd  8523