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 Description: A number commutes with its additive inverse. Compare remulinvcom 39946. (Contributed by SN, 5-May-2024.)
Hypotheses
Ref Expression
addinvcom.1 (𝜑 → (𝐴 + 𝐵) = 0)
Assertion
Ref Expression
addinvcom (𝜑 → (𝐵 + 𝐴) = 0)

Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssidd 3917 . . . . 5 (𝜑 → ℂ ⊆ ℂ)
2 simpl 486 . . . . . . 7 (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) → (𝐴 + 𝑥) = 0)
32rgenw 3082 . . . . . 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 39930 . . . . . 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 10684 . . . . . 6 0 ∈ ℂ
9 sn-subeu 39940 . . . . . 6 ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
105, 8, 9sylancl 589 . . . . 5 (𝜑 → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
11 riotass2 7144 . . . . 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 7164 . . . . . . . 8 (𝑥 = 𝐵 → (𝐴 + 𝑥) = (𝐴 + 𝐵))
1615eqeq1d 2760 . . . . . . 7 (𝑥 = 𝐵 → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + 𝐵) = 0))
1716riota2 7139 . . . . . 6 ((𝐵 ∈ ℂ ∧ ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) → ((𝐴 + 𝐵) = 0 ↔ (𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵))
1814, 10, 17syl2anc 587 . . . . 5 (𝜑 → ((𝐴 + 𝐵) = 0 ↔ (𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵))
1913, 18mpbid 235 . . . 4 (𝜑 → (𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵)
2012, 19eqtrd 2793 . . 3 (𝜑 → (𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)) = 𝐵)
21 reurmo 3343 . . . . . 6 (∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0 → ∃*𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
222rmoimi 3658 . . . . . 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 3340 . . . . 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 586 . . . 4 (𝜑 → ∃!𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0))
26 oveq1 7163 . . . . . . 7 (𝑥 = 𝐵 → (𝑥 + 𝐴) = (𝐵 + 𝐴))
2726eqeq1d 2760 . . . . . 6 (𝑥 = 𝐵 → ((𝑥 + 𝐴) = 0 ↔ (𝐵 + 𝐴) = 0))
2816, 27anbi12d 633 . . . . 5 (𝑥 = 𝐵 → (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) ↔ ((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0)))
2928riota2 7139 . . . 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 587 . . 3 (𝜑 → (((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0) ↔ (𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)) = 𝐵))
3120, 30mpbird 260 . 2 (𝜑 → ((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0))
3231simprd 499 1 (𝜑 → (𝐵 + 𝐴) = 0)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  ∃!wreu 3072  ∃*wrmo 3073   ⊆ wss 3860  ℩crio 7113  (class class class)co 7156  ℂcc 10586  0cc0 10588   + caddc 10591 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-po 5447  df-so 5448  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-er 8305  df-en 8541  df-dom 8542  df-sdom 8543  df-pnf 10728  df-mnf 10729  df-ltxr 10731  df-2 11750  df-3 11751  df-resub 39881 This theorem is referenced by: (None)
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