Theorem addinvcom 42464

Description: A number commutes with its additive inverse. Compare remulinvcom 42465. (Contributed by SN, 5-May-2024.)

Hypotheses

Ref	Expression
addinvcom.a	⊢ (𝜑 → 𝐴 ∈ ℂ)
addinvcom.b	⊢ (𝜑 → 𝐵 ∈ ℂ)
addinvcom.1	⊢ (𝜑 → (𝐴 + 𝐵) = 0)

Assertion

Ref	Expression
addinvcom	⊢ (𝜑 → (𝐵 + 𝐴) = 0)

Proof of Theorem addinvcom

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssidd 3958	. . . . 5 ⊢ (𝜑 → ℂ ⊆ ℂ)
2		simpl 482	. . . . . . 7 ⊢ (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) → (𝐴 + 𝑥) = 0)
3	2	rgenw 3051	. . . . . 6 ⊢ ∀𝑥 ∈ ℂ (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) → (𝐴 + 𝑥) = 0)
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ℂ (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) → (𝐴 + 𝑥) = 0))
5		addinvcom.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
6		sn-negex12 42449	. . . . . 6 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0))
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0))
8		0cn 11101	. . . . . 6 ⊢ 0 ∈ ℂ
9		sn-subeu 42459	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
10	5, 8, 9	sylancl 586	. . . . 5 ⊢ (𝜑 → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
11		riotass2 7333	. . . . 5 ⊢ (((ℂ ⊆ ℂ ∧ ∀𝑥 ∈ ℂ (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) → (𝐴 + 𝑥) = 0)) ∧ (∃𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) ∧ ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)) → (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)) = (℩𝑥 ∈ ℂ (𝐴 + 𝑥) = 0))
12	1, 4, 7, 10, 11	syl22anc 838	. . . 4 ⊢ (𝜑 → (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)) = (℩𝑥 ∈ ℂ (𝐴 + 𝑥) = 0))
13		addinvcom.1	. . . . 5 ⊢ (𝜑 → (𝐴 + 𝐵) = 0)
14		addinvcom.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ)
15		oveq2 7354	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐴 + 𝑥) = (𝐴 + 𝐵))
16	15	eqeq1d 2733	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + 𝐵) = 0))
17	16	riota2 7328	. . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) → ((𝐴 + 𝐵) = 0 ↔ (℩𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵))
18	14, 10, 17	syl2anc 584	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐵) = 0 ↔ (℩𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵))
19	13, 18	mpbid 232	. . . 4 ⊢ (𝜑 → (℩𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵)
20	12, 19	eqtrd 2766	. . 3 ⊢ (𝜑 → (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)) = 𝐵)
21		reurmo 3349	. . . . . 6 ⊢ (∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0 → ∃*𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
22	2	rmoimi 3701	. . . . . 6 ⊢ (∃*𝑥 ∈ ℂ (𝐴 + 𝑥) = 0 → ∃*𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0))
23	10, 21, 22	3syl 18	. . . . 5 ⊢ (𝜑 → ∃*𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0))
24		reu5 3348	. . . . 5 ⊢ (∃!𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) ↔ (∃𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) ∧ ∃*𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)))
25	7, 23, 24	sylanbrc 583	. . . 4 ⊢ (𝜑 → ∃!𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0))
26		oveq1 7353	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 + 𝐴) = (𝐵 + 𝐴))
27	26	eqeq1d 2733	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑥 + 𝐴) = 0 ↔ (𝐵 + 𝐴) = 0))
28	16, 27	anbi12d 632	. . . . 5 ⊢ (𝑥 = 𝐵 → (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) ↔ ((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0)))
29	28	riota2 7328	. . . 4 ⊢ ((𝐵 ∈ ℂ ∧ ∃!𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)) → (((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0) ↔ (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)) = 𝐵))
30	14, 25, 29	syl2anc 584	. . 3 ⊢ (𝜑 → (((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0) ↔ (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)) = 𝐵))
31	20, 30	mpbird 257	. 2 ⊢ (𝜑 → ((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0))
32	31	simprd 495	1 ⊢ (𝜑 → (𝐵 + 𝐴) = 0)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  ∃!wreu 3344  ∃*wrmo 3345   ⊆ wss 3902  ℩crio 7302  (class class class)co 7346  ℂcc 11001  0cc0 11003   + caddc 11006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-po 5524  df-so 5525  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11145  df-mnf 11146  df-ltxr 11148  df-2 12185  df-3 12186  df-resub 42398

This theorem is referenced by: (None)