Theorem addinvcom 40334

Description: A number commutes with its additive inverse. Compare remulinvcom 40335. (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 3940	. . . . 5 ⊢ (𝜑 → ℂ ⊆ ℂ)
2		simpl 482	. . . . . . 7 ⊢ (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) → (𝐴 + 𝑥) = 0)
3	2	rgenw 3075	. . . . . 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 40319	. . . . . 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 10898	. . . . . 6 ⊢ 0 ∈ ℂ
9		sn-subeu 40329	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
10	5, 8, 9	sylancl 585	. . . . 5 ⊢ (𝜑 → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
11		riotass2 7243	. . . . 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 835	. . . 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 7263	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐴 + 𝑥) = (𝐴 + 𝐵))
16	15	eqeq1d 2740	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + 𝐵) = 0))
17	16	riota2 7238	. . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) → ((𝐴 + 𝐵) = 0 ↔ (℩𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵))
18	14, 10, 17	syl2anc 583	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐵) = 0 ↔ (℩𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵))
19	13, 18	mpbid 231	. . . 4 ⊢ (𝜑 → (℩𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵)
20	12, 19	eqtrd 2778	. . 3 ⊢ (𝜑 → (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)) = 𝐵)
21		reurmo 3354	. . . . . 6 ⊢ (∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0 → ∃*𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
22	2	rmoimi 3672	. . . . . 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 3351	. . . . 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 582	. . . 4 ⊢ (𝜑 → ∃!𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0))
26		oveq1 7262	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 + 𝐴) = (𝐵 + 𝐴))
27	26	eqeq1d 2740	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑥 + 𝐴) = 0 ↔ (𝐵 + 𝐴) = 0))
28	16, 27	anbi12d 630	. . . . 5 ⊢ (𝑥 = 𝐵 → (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) ↔ ((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0)))
29	28	riota2 7238	. . . 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 583	. . 3 ⊢ (𝜑 → (((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0) ↔ (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)) = 𝐵))
31	20, 30	mpbird 256	. 2 ⊢ (𝜑 → ((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0))
32	31	simprd 495	1 ⊢ (𝜑 → (𝐵 + 𝐴) = 0)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  ∃!wreu 3065  ∃*wrmo 3066   ⊆ wss 3883  ℩crio 7211  (class class class)co 7255  ℂcc 10800  0cc0 10802   + caddc 10805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-ltxr 10945  df-2 11966  df-3 11967  df-resub 40270

This theorem is referenced by: (None)