Theorem addinvcom 42427

Description: A number commutes with its additive inverse. Compare remulinvcom 42428. (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 3973	. . . . 5 ⊢ (𝜑 → ℂ ⊆ ℂ)
2		simpl 482	. . . . . . 7 ⊢ (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) → (𝐴 + 𝑥) = 0)
3	2	rgenw 3049	. . . . . 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 42412	. . . . . 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 11173	. . . . . 6 ⊢ 0 ∈ ℂ
9		sn-subeu 42422	. . . . . 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 7377	. . . . 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 7398	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐴 + 𝑥) = (𝐴 + 𝐵))
16	15	eqeq1d 2732	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + 𝐵) = 0))
17	16	riota2 7372	. . . . . 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 2765	. . 3 ⊢ (𝜑 → (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)) = 𝐵)
21		reurmo 3359	. . . . . 6 ⊢ (∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0 → ∃*𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
22	2	rmoimi 3716	. . . . . 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 3358	. . . . 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 7397	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 + 𝐴) = (𝐵 + 𝐴))
27	26	eqeq1d 2732	. . . . . 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 7372	. . . 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 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  ∃!wreu 3354  ∃*wrmo 3355   ⊆ wss 3917  ℩crio 7346  (class class class)co 7390  ℂcc 11073  0cc0 11075   + caddc 11078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-po 5549  df-so 5550  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-ltxr 11220  df-2 12256  df-3 12257  df-resub 42361

This theorem is referenced by: (None)