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Theorem addinvcom 41386
Description: A number commutes with its additive inverse. Compare remulinvcom 41387. (Contributed by SN, 5-May-2024.)
Hypotheses
Ref Expression
addinvcom.a (𝜑𝐴 ∈ ℂ)
addinvcom.b (𝜑𝐵 ∈ 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.
StepHypRef Expression
1 ssidd 4005 . . . . 5 (𝜑 → ℂ ⊆ ℂ)
2 simpl 483 . . . . . . 7 (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) → (𝐴 + 𝑥) = 0)
32rgenw 3065 . . . . . 6 𝑥 ∈ ℂ (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) → (𝐴 + 𝑥) = 0)
43a1i 11 . . . . 5 (𝜑 → ∀𝑥 ∈ ℂ (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) → (𝐴 + 𝑥) = 0))
5 addinvcom.a . . . . . 6 (𝜑𝐴 ∈ ℂ)
6 sn-negex12 41371 . . . . . 6 (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0))
75, 6syl 17 . . . . 5 (𝜑 → ∃𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0))
8 0cn 11208 . . . . . 6 0 ∈ ℂ
9 sn-subeu 41381 . . . . . 6 ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
105, 8, 9sylancl 586 . . . . 5 (𝜑 → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
11 riotass2 7398 . . . . 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))
121, 4, 7, 10, 11syl22anc 837 . . . 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 (𝜑𝐵 ∈ ℂ)
15 oveq2 7419 . . . . . . . 8 (𝑥 = 𝐵 → (𝐴 + 𝑥) = (𝐴 + 𝐵))
1615eqeq1d 2734 . . . . . . 7 (𝑥 = 𝐵 → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + 𝐵) = 0))
1716riota2 7393 . . . . . 6 ((𝐵 ∈ ℂ ∧ ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) → ((𝐴 + 𝐵) = 0 ↔ (𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵))
1814, 10, 17syl2anc 584 . . . . 5 (𝜑 → ((𝐴 + 𝐵) = 0 ↔ (𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵))
1913, 18mpbid 231 . . . 4 (𝜑 → (𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵)
2012, 19eqtrd 2772 . . 3 (𝜑 → (𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)) = 𝐵)
21 reurmo 3379 . . . . . 6 (∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0 → ∃*𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
222rmoimi 3738 . . . . . 6 (∃*𝑥 ∈ ℂ (𝐴 + 𝑥) = 0 → ∃*𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0))
2310, 21, 223syl 18 . . . . 5 (𝜑 → ∃*𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0))
24 reu5 3378 . . . . 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)))
257, 23, 24sylanbrc 583 . . . 4 (𝜑 → ∃!𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0))
26 oveq1 7418 . . . . . . 7 (𝑥 = 𝐵 → (𝑥 + 𝐴) = (𝐵 + 𝐴))
2726eqeq1d 2734 . . . . . 6 (𝑥 = 𝐵 → ((𝑥 + 𝐴) = 0 ↔ (𝐵 + 𝐴) = 0))
2816, 27anbi12d 631 . . . . 5 (𝑥 = 𝐵 → (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) ↔ ((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0)))
2928riota2 7393 . . . 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))
3014, 25, 29syl2anc 584 . . 3 (𝜑 → (((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0) ↔ (𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)) = 𝐵))
3120, 30mpbird 256 . 2 (𝜑 → ((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0))
3231simprd 496 1 (𝜑 → (𝐵 + 𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3061  wrex 3070  ∃!wreu 3374  ∃*wrmo 3375  wss 3948  crio 7366  (class class class)co 7411  cc 11110  0cc0 11112   + caddc 11115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11252  df-mnf 11253  df-ltxr 11255  df-2 12277  df-3 12278  df-resub 41321
This theorem is referenced by: (None)
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