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Mirrors > Home > MPE Home > Th. List > cnegex2 | Structured version Visualization version GIF version |
Description: Existence of a left inverse for addition. (Contributed by Scott Fenton, 3-Jan-2013.) |
Ref | Expression |
---|---|
cnegex2 | ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-icn 10590 | . . . 4 ⊢ i ∈ ℂ | |
2 | 1, 1 | mulcli 10642 | . . 3 ⊢ (i · i) ∈ ℂ |
3 | mulcl 10615 | . . 3 ⊢ (((i · i) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((i · i) · 𝐴) ∈ ℂ) | |
4 | 2, 3 | mpan 688 | . 2 ⊢ (𝐴 ∈ ℂ → ((i · i) · 𝐴) ∈ ℂ) |
5 | mulid2 10634 | . . . 4 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
6 | 5 | oveq2d 7166 | . . 3 ⊢ (𝐴 ∈ ℂ → (((i · i) · 𝐴) + (1 · 𝐴)) = (((i · i) · 𝐴) + 𝐴)) |
7 | ax-i2m1 10599 | . . . . 5 ⊢ ((i · i) + 1) = 0 | |
8 | 7 | oveq1i 7160 | . . . 4 ⊢ (((i · i) + 1) · 𝐴) = (0 · 𝐴) |
9 | ax-1cn 10589 | . . . . 5 ⊢ 1 ∈ ℂ | |
10 | adddir 10626 | . . . . 5 ⊢ (((i · i) ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (((i · i) + 1) · 𝐴) = (((i · i) · 𝐴) + (1 · 𝐴))) | |
11 | 2, 9, 10 | mp3an12 1447 | . . . 4 ⊢ (𝐴 ∈ ℂ → (((i · i) + 1) · 𝐴) = (((i · i) · 𝐴) + (1 · 𝐴))) |
12 | mul02 10812 | . . . 4 ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) | |
13 | 8, 11, 12 | 3eqtr3a 2880 | . . 3 ⊢ (𝐴 ∈ ℂ → (((i · i) · 𝐴) + (1 · 𝐴)) = 0) |
14 | 6, 13 | eqtr3d 2858 | . 2 ⊢ (𝐴 ∈ ℂ → (((i · i) · 𝐴) + 𝐴) = 0) |
15 | oveq1 7157 | . . . 4 ⊢ (𝑥 = ((i · i) · 𝐴) → (𝑥 + 𝐴) = (((i · i) · 𝐴) + 𝐴)) | |
16 | 15 | eqeq1d 2823 | . . 3 ⊢ (𝑥 = ((i · i) · 𝐴) → ((𝑥 + 𝐴) = 0 ↔ (((i · i) · 𝐴) + 𝐴) = 0)) |
17 | 16 | rspcev 3623 | . 2 ⊢ ((((i · i) · 𝐴) ∈ ℂ ∧ (((i · i) · 𝐴) + 𝐴) = 0) → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0) |
18 | 4, 14, 17 | syl2anc 586 | 1 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 (class class class)co 7150 ℂcc 10529 0cc0 10531 1c1 10532 ici 10533 + caddc 10534 · cmul 10536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 |
This theorem is referenced by: addcan 10818 |
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