Theorem subval 11418

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		eqeq2 2773	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑧 + 𝑥) = 𝑦 ↔ (𝑧 + 𝑥) = 𝐴))
2	1	riotabidv 7351	. 2 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝑦) = (℩𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝐴))
3		oveq1 7399	. . . 4 ⊢ (𝑧 = 𝐵 → (𝑧 + 𝑥) = (𝐵 + 𝑥))
4	3	eqeq1d 2763	. . 3 ⊢ (𝑧 = 𝐵 → ((𝑧 + 𝑥) = 𝐴 ↔ (𝐵 + 𝑥) = 𝐴))
5	4	riotabidv 7351	. 2 ⊢ (𝑧 = 𝐵 → (℩𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝐴) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))
6		df-sub 11413	. 2 ⊢ − = (𝑦 ∈ ℂ, 𝑧 ∈ ℂ ↦ (℩𝑥 ∈ ℂ (𝑧 + 𝑥) = 𝑦))
7		riotaex 7353	. 2 ⊢ (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) ∈ V
8	2, 5, 6, 7	ovmpo 7552	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ℩crio 7348  (class class class)co 7392  ℂcc 11068   + caddc 11073   − cmin 11411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-sub 11413

This theorem is referenced by:  subcl  11426  subf  11429  subadd  11430  sn-subcl  43001  sn-subf  43002  resubeqsub  43003