Theorem subval 8376

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 8375	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)
2		riotacl 5992	. . . 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 2240	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑧 + 𝑥) = 𝑦 ↔ (𝑧 + 𝑥) = 𝐴))
6	5	riotabidv 5978	. . 3 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝑦) = (℩𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝐴))
7		oveq1 6030	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑧 + 𝑥) = (𝐵 + 𝑥))
8	7	eqeq1d 2239	. . . 4 ⊢ (𝑧 = 𝐵 → ((𝑧 + 𝑥) = 𝐴 ↔ (𝐵 + 𝑥) = 𝐴))
9	8	riotabidv 5978	. . 3 ⊢ (𝑧 = 𝐵 → (℩𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝐴) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
10		df-sub 8357	. . 3 ⊢ − = (𝑦 ∈ ℂ, 𝑧 ∈ ℂ ↦ (℩𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝑦))
11	6, 9, 10	ovmpog 6161	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ ℂ) → (𝐴 − 𝐵) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
12	4, 11	mpd3an3 1374	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2201  ∃!wreu 2511  ℩crio 5975  (class class class)co 6023  ℂcc 8035   + caddc 8040   − cmin 8355

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-setind 4637  ax-resscn 8129  ax-1cn 8130  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-addass 8139  ax-distr 8141  ax-i2m1 8142  ax-0id 8145  ax-rnegex 8146  ax-cnre 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-iota 5288  df-fun 5330  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-sub 8357

This theorem is referenced by:  subcl  8383  subf  8386  subadd  8387