Theorem subval 8306

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 8305	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)
2		riotacl 5943	. . . 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 2219	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑧 + 𝑥) = 𝑦 ↔ (𝑧 + 𝑥) = 𝐴))
6	5	riotabidv 5929	. . 3 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝑦) = (℩𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝐴))
7		oveq1 5981	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑧 + 𝑥) = (𝐵 + 𝑥))
8	7	eqeq1d 2218	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝑧 + 𝑥) = 𝐴 ↔ (𝐵 + 𝑥) = 𝐴))
9	8	riotabidv 5929	. . 3 ⊢ (𝑧 = 𝐵 → (℩𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝐴) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
10		df-sub 8287	. . 3 ⊢ − = (𝑦 ∈ ℂ, 𝑧 ∈ ℂ ↦ (℩𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝑦))
11	6, 9, 10	ovmpog 6110	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ ℂ) → (𝐴 − 𝐵) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
12	4, 11	mpd3an3 1353	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1375   ∈ wcel 2180  ∃!wreu 2490  ℩crio 5926  (class class class)co 5974  ℂcc 7965   + caddc 7970   − cmin 8285

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-setind 4606  ax-resscn 8059  ax-1cn 8060  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-addass 8069  ax-distr 8071  ax-i2m1 8072  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-br 4063  df-opab 4125  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-iota 5254  df-fun 5296  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-sub 8287

This theorem is referenced by:  subcl  8313  subf  8316  subadd  8317