Theorem addinvcom 42916

Description: A number commutes with its additive inverse. Compare remulinvcom 42917. (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 3945	. . . . 5 ⊢ (𝜑 → ℂ ⊆ ℂ)
2		simpl 483	. . . . . . 7 ⊢ (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) → (𝐴 + 𝑥) = 0)
3	2	rgenw 3058	. . . . . 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 42901	. . . . . 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 11134	. . . . . 6 ⊢ 0 ∈ ℂ
9		sn-subeu 42911	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
10	5, 8, 9	sylancl 592	. . . . 5 ⊢ (𝜑 → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
11		riotass2 7350	. . . . 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 844	. . . 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 7371	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐴 + 𝑥) = (𝐴 + 𝐵))
16	15	eqeq1d 2742	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + 𝐵) = 0))
17	16	riota2 7345	. . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) → ((𝐴 + 𝐵) = 0 ↔ (℩𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵))
18	14, 10, 17	syl2anc 590	. . . . 5 ⊢ (𝜑 → ((𝐴 + 𝐵) = 0 ↔ (℩𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵))
19	13, 18	mpbid 233	. . . 4 ⊢ (𝜑 → (℩𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵)
20	12, 19	eqtrd 2775	. . 3 ⊢ (𝜑 → (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)) = 𝐵)
21		reurmo 3348	. . . . . 6 ⊢ (∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0 → ∃*𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
22	2	rmoimi 3690	. . . . . 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 3347	. . . . 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 589	. . . 4 ⊢ (𝜑 → ∃!𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0))
26		oveq1 7370	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥 + 𝐴) = (𝐵 + 𝐴))
27	26	eqeq1d 2742	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑥 + 𝐴) = 0 ↔ (𝐵 + 𝐴) = 0))
28	16, 27	anbi12d 638	. . . . 5 ⊢ (𝑥 = 𝐵 → (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) ↔ ((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0)))
29	28	riota2 7345	. . . 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 590	. . 3 ⊢ (𝜑 → (((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0) ↔ (℩𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)) = 𝐵))
31	20, 30	mpbird 258	. 2 ⊢ (𝜑 → ((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0))
32	31	simprd 496	1 ⊢ (𝜑 → (𝐵 + 𝐴) = 0)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  ∃!wreu 3343  ∃*wrmo 3344   ⊆ wss 3890  ℩crio 7319  (class class class)co 7363  ℂcc 11034  0cc0 11036   + caddc 11039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-po 5533  df-so 5534  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-pnf 11179  df-mnf 11180  df-ltxr 11182  df-2 12242  df-3 12243  df-resub 42850

This theorem is referenced by: (None)