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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 9874 | . . . 4 ⊢ i ∈ ℂ | |
2 | 1, 1 | mulcli 9924 | . . 3 ⊢ (i · i) ∈ ℂ |
3 | mulcl 9899 | . . 3 ⊢ (((i · i) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((i · i) · 𝐴) ∈ ℂ) | |
4 | 2, 3 | mpan 702 | . 2 ⊢ (𝐴 ∈ ℂ → ((i · i) · 𝐴) ∈ ℂ) |
5 | mulid2 9917 | . . . 4 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
6 | 5 | oveq2d 6565 | . . 3 ⊢ (𝐴 ∈ ℂ → (((i · i) · 𝐴) + (1 · 𝐴)) = (((i · i) · 𝐴) + 𝐴)) |
7 | ax-i2m1 9883 | . . . . 5 ⊢ ((i · i) + 1) = 0 | |
8 | 7 | oveq1i 6559 | . . . 4 ⊢ (((i · i) + 1) · 𝐴) = (0 · 𝐴) |
9 | ax-1cn 9873 | . . . . 5 ⊢ 1 ∈ ℂ | |
10 | adddir 9910 | . . . . 5 ⊢ (((i · i) ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (((i · i) + 1) · 𝐴) = (((i · i) · 𝐴) + (1 · 𝐴))) | |
11 | 2, 9, 10 | mp3an12 1406 | . . . 4 ⊢ (𝐴 ∈ ℂ → (((i · i) + 1) · 𝐴) = (((i · i) · 𝐴) + (1 · 𝐴))) |
12 | mul02 10093 | . . . 4 ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) | |
13 | 8, 11, 12 | 3eqtr3a 2668 | . . 3 ⊢ (𝐴 ∈ ℂ → (((i · i) · 𝐴) + (1 · 𝐴)) = 0) |
14 | 6, 13 | eqtr3d 2646 | . 2 ⊢ (𝐴 ∈ ℂ → (((i · i) · 𝐴) + 𝐴) = 0) |
15 | oveq1 6556 | . . . 4 ⊢ (𝑥 = ((i · i) · 𝐴) → (𝑥 + 𝐴) = (((i · i) · 𝐴) + 𝐴)) | |
16 | 15 | eqeq1d 2612 | . . 3 ⊢ (𝑥 = ((i · i) · 𝐴) → ((𝑥 + 𝐴) = 0 ↔ (((i · i) · 𝐴) + 𝐴) = 0)) |
17 | 16 | rspcev 3282 | . 2 ⊢ ((((i · i) · 𝐴) ∈ ℂ ∧ (((i · i) · 𝐴) + 𝐴) = 0) → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0) |
18 | 4, 14, 17 | syl2anc 691 | 1 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 ici 9817 + caddc 9818 · cmul 9820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 |
This theorem is referenced by: addcan 10099 |
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