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Mirrors > Home > MPE Home > Th. List > Mathboxes > ipiiie0 | Structured version Visualization version GIF version |
Description: The multiplicative inverse of i (per i4 13919) is also its additive inverse. (Contributed by SN, 30-Jun-2024.) |
Ref | Expression |
---|---|
ipiiie0 | ⊢ (i + (i · (i · i))) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | it1ei 40415 | . . . 4 ⊢ (i · 1) = i | |
2 | 1 | eqcomi 2747 | . . 3 ⊢ i = (i · 1) |
3 | reixi 40401 | . . . 4 ⊢ (i · i) = (0 −ℝ 1) | |
4 | 3 | oveq2i 7288 | . . 3 ⊢ (i · (i · i)) = (i · (0 −ℝ 1)) |
5 | 2, 4 | oveq12i 7289 | . 2 ⊢ (i + (i · (i · i))) = ((i · 1) + (i · (0 −ℝ 1))) |
6 | ax-icn 10928 | . . 3 ⊢ i ∈ ℂ | |
7 | ax-1cn 10927 | . . 3 ⊢ 1 ∈ ℂ | |
8 | 1re 10973 | . . . . 5 ⊢ 1 ∈ ℝ | |
9 | rernegcl 40351 | . . . . 5 ⊢ (1 ∈ ℝ → (0 −ℝ 1) ∈ ℝ) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ (0 −ℝ 1) ∈ ℝ |
11 | 10 | recni 10987 | . . 3 ⊢ (0 −ℝ 1) ∈ ℂ |
12 | 6, 7, 11 | adddii 10985 | . 2 ⊢ (i · (1 + (0 −ℝ 1))) = ((i · 1) + (i · (0 −ℝ 1))) |
13 | renegid 40353 | . . . . 5 ⊢ (1 ∈ ℝ → (1 + (0 −ℝ 1)) = 0) | |
14 | 8, 13 | ax-mp 5 | . . . 4 ⊢ (1 + (0 −ℝ 1)) = 0 |
15 | 14 | oveq2i 7288 | . . 3 ⊢ (i · (1 + (0 −ℝ 1))) = (i · 0) |
16 | sn-it0e0 40394 | . . 3 ⊢ (i · 0) = 0 | |
17 | 15, 16 | eqtri 2766 | . 2 ⊢ (i · (1 + (0 −ℝ 1))) = 0 |
18 | 5, 12, 17 | 3eqtr2i 2772 | 1 ⊢ (i + (i · (i · i))) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2106 (class class class)co 7277 ℝcr 10868 0cc0 10869 1c1 10870 ici 10871 + caddc 10872 · cmul 10874 −ℝ cresub 40345 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-br 5077 df-opab 5139 df-mpt 5160 df-id 5491 df-po 5505 df-so 5506 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-pnf 11009 df-mnf 11010 df-ltxr 11012 df-2 12034 df-3 12035 df-resub 40346 |
This theorem is referenced by: sn-0tie0 40418 sn-inelr 40432 |
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