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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 10442 | . . . 4 ⊢ i ∈ ℂ | |
2 | 1, 1 | mulcli 10494 | . . 3 ⊢ (i · i) ∈ ℂ |
3 | mulcl 10467 | . . 3 ⊢ (((i · i) ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((i · i) · 𝐴) ∈ ℂ) | |
4 | 2, 3 | mpan 686 | . 2 ⊢ (𝐴 ∈ ℂ → ((i · i) · 𝐴) ∈ ℂ) |
5 | mulid2 10486 | . . . 4 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
6 | 5 | oveq2d 7032 | . . 3 ⊢ (𝐴 ∈ ℂ → (((i · i) · 𝐴) + (1 · 𝐴)) = (((i · i) · 𝐴) + 𝐴)) |
7 | ax-i2m1 10451 | . . . . 5 ⊢ ((i · i) + 1) = 0 | |
8 | 7 | oveq1i 7026 | . . . 4 ⊢ (((i · i) + 1) · 𝐴) = (0 · 𝐴) |
9 | ax-1cn 10441 | . . . . 5 ⊢ 1 ∈ ℂ | |
10 | adddir 10478 | . . . . 5 ⊢ (((i · i) ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (((i · i) + 1) · 𝐴) = (((i · i) · 𝐴) + (1 · 𝐴))) | |
11 | 2, 9, 10 | mp3an12 1443 | . . . 4 ⊢ (𝐴 ∈ ℂ → (((i · i) + 1) · 𝐴) = (((i · i) · 𝐴) + (1 · 𝐴))) |
12 | mul02 10665 | . . . 4 ⊢ (𝐴 ∈ ℂ → (0 · 𝐴) = 0) | |
13 | 8, 11, 12 | 3eqtr3a 2855 | . . 3 ⊢ (𝐴 ∈ ℂ → (((i · i) · 𝐴) + (1 · 𝐴)) = 0) |
14 | 6, 13 | eqtr3d 2833 | . 2 ⊢ (𝐴 ∈ ℂ → (((i · i) · 𝐴) + 𝐴) = 0) |
15 | oveq1 7023 | . . . 4 ⊢ (𝑥 = ((i · i) · 𝐴) → (𝑥 + 𝐴) = (((i · i) · 𝐴) + 𝐴)) | |
16 | 15 | eqeq1d 2797 | . . 3 ⊢ (𝑥 = ((i · i) · 𝐴) → ((𝑥 + 𝐴) = 0 ↔ (((i · i) · 𝐴) + 𝐴) = 0)) |
17 | 16 | rspcev 3559 | . 2 ⊢ ((((i · i) · 𝐴) ∈ ℂ ∧ (((i · i) · 𝐴) + 𝐴) = 0) → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0) |
18 | 4, 14, 17 | syl2anc 584 | 1 ⊢ (𝐴 ∈ ℂ → ∃𝑥 ∈ ℂ (𝑥 + 𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 ∃wrex 3106 (class class class)co 7016 ℂcc 10381 0cc0 10383 1c1 10384 ici 10385 + caddc 10386 · cmul 10388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-po 5362 df-so 5363 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-ov 7019 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-ltxr 10526 |
This theorem is referenced by: addcan 10671 |
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