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Theorem addinvcom 41186
Description: A number commutes with its additive inverse. Compare remulinvcom 41187. (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 4003 . . . . 5 (𝜑 → ℂ ⊆ ℂ)
2 simpl 484 . . . . . . 7 (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) → (𝐴 + 𝑥) = 0)
32rgenw 3066 . . . . . 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 41171 . . . . . 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 11193 . . . . . 6 0 ∈ ℂ
9 sn-subeu 41181 . . . . . 6 ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
105, 8, 9sylancl 587 . . . . 5 (𝜑 → ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
11 riotass2 7383 . . . . 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 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 (𝜑𝐵 ∈ ℂ)
15 oveq2 7404 . . . . . . . 8 (𝑥 = 𝐵 → (𝐴 + 𝑥) = (𝐴 + 𝐵))
1615eqeq1d 2735 . . . . . . 7 (𝑥 = 𝐵 → ((𝐴 + 𝑥) = 0 ↔ (𝐴 + 𝐵) = 0))
1716riota2 7378 . . . . . 6 ((𝐵 ∈ ℂ ∧ ∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) → ((𝐴 + 𝐵) = 0 ↔ (𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵))
1814, 10, 17syl2anc 585 . . . . 5 (𝜑 → ((𝐴 + 𝐵) = 0 ↔ (𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵))
1913, 18mpbid 231 . . . 4 (𝜑 → (𝑥 ∈ ℂ (𝐴 + 𝑥) = 0) = 𝐵)
2012, 19eqtrd 2773 . . 3 (𝜑 → (𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)) = 𝐵)
21 reurmo 3380 . . . . . 6 (∃!𝑥 ∈ ℂ (𝐴 + 𝑥) = 0 → ∃*𝑥 ∈ ℂ (𝐴 + 𝑥) = 0)
222rmoimi 3736 . . . . . 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 3379 . . . . 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 584 . . . 4 (𝜑 → ∃!𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0))
26 oveq1 7403 . . . . . . 7 (𝑥 = 𝐵 → (𝑥 + 𝐴) = (𝐵 + 𝐴))
2726eqeq1d 2735 . . . . . 6 (𝑥 = 𝐵 → ((𝑥 + 𝐴) = 0 ↔ (𝐵 + 𝐴) = 0))
2816, 27anbi12d 632 . . . . 5 (𝑥 = 𝐵 → (((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0) ↔ ((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0)))
2928riota2 7378 . . . 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 585 . . 3 (𝜑 → (((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0) ↔ (𝑥 ∈ ℂ ((𝐴 + 𝑥) = 0 ∧ (𝑥 + 𝐴) = 0)) = 𝐵))
3120, 30mpbird 257 . 2 (𝜑 → ((𝐴 + 𝐵) = 0 ∧ (𝐵 + 𝐴) = 0))
3231simprd 497 1 (𝜑 → (𝐵 + 𝐴) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  ∃!wreu 3375  ∃*wrmo 3376  wss 3946  crio 7351  (class class class)co 7396  cc 11095  0cc0 11097   + caddc 11100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712  ax-resscn 11154  ax-1cn 11155  ax-icn 11156  ax-addcl 11157  ax-addrcl 11158  ax-mulcl 11159  ax-mulrcl 11160  ax-addass 11162  ax-mulass 11163  ax-distr 11164  ax-i2m1 11165  ax-1ne0 11166  ax-1rid 11167  ax-rnegex 11168  ax-rrecex 11169  ax-cnre 11170  ax-pre-lttri 11171  ax-pre-lttrn 11172  ax-pre-ltadd 11173
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-po 5584  df-so 5585  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-riota 7352  df-ov 7399  df-oprab 7400  df-mpo 7401  df-er 8691  df-en 8928  df-dom 8929  df-sdom 8930  df-pnf 11237  df-mnf 11238  df-ltxr 11240  df-2 12262  df-3 12263  df-resub 41121
This theorem is referenced by: (None)
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