Theorem addinvcom 42407

Description: A number commutes with its additive inverse. Compare remulinvcom 42408. (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 4032	. . . . 5 ⊢ (𝜑 → ℂ ⊆ ℂ)
2		simpl 482	. . . . . . 7 ⊢ (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) → (𝐴 + 𝑥) = 0)
3	2	rgenw 3071	. . . . . 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 42392	. . . . . 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 11282	. . . . . 6 ⊢ 0 ∈ ℂ
9		sn-subeu 42402	. . . . . 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 7435	. . . . 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 7456	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐴 + 𝑥) = (𝐴 + 𝐵))
16	15	eqeq1d 2742	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + 𝐵) = 0))
17	16	riota2 7430	. . . . . 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 232	. . . 4 ⊢ (𝜑 → (℩𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵)
20	12, 19	eqtrd 2780	. . 3 ⊢ (𝜑 → (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)) = 𝐵)
21		reurmo 3391	. . . . . 6 ⊢ (∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0 → ∃*𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
22	2	rmoimi 3764	. . . . . 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 3390	. . . . 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 7455	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 + 𝐴) = (𝐵 + 𝐴))
27	26	eqeq1d 2742	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑥 + 𝐴) = 0 ↔ (𝐵 + 𝐴) = 0))
28	16, 27	anbi12d 631	. . . . 5 ⊢ (𝑥 = 𝐵 → (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) ↔ ((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0)))
29	28	riota2 7430	. . . 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 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 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  ∃!wreu 3386  ∃*wrmo 3387   ⊆ wss 3976  ℩crio 7403  (class class class)co 7448  ℂcc 11182  0cc0 11184   + caddc 11187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-ltxr 11329  df-2 12356  df-3 12357  df-resub 42342

This theorem is referenced by: (None)