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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 13738) 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 40067 | . . . 4 ⊢ (i · 1) = i | |
2 | 1 | eqcomi 2745 | . . 3 ⊢ i = (i · 1) |
3 | reixi 40053 | . . . 4 ⊢ (i · i) = (0 −ℝ 1) | |
4 | 3 | oveq2i 7202 | . . 3 ⊢ (i · (i · i)) = (i · (0 −ℝ 1)) |
5 | 2, 4 | oveq12i 7203 | . 2 ⊢ (i + (i · (i · i))) = ((i · 1) + (i · (0 −ℝ 1))) |
6 | ax-icn 10753 | . . 3 ⊢ i ∈ ℂ | |
7 | ax-1cn 10752 | . . 3 ⊢ 1 ∈ ℂ | |
8 | 1re 10798 | . . . . 5 ⊢ 1 ∈ ℝ | |
9 | rernegcl 40003 | . . . . 5 ⊢ (1 ∈ ℝ → (0 −ℝ 1) ∈ ℝ) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ (0 −ℝ 1) ∈ ℝ |
11 | 10 | recni 10812 | . . 3 ⊢ (0 −ℝ 1) ∈ ℂ |
12 | 6, 7, 11 | adddii 10810 | . 2 ⊢ (i · (1 + (0 −ℝ 1))) = ((i · 1) + (i · (0 −ℝ 1))) |
13 | renegid 40005 | . . . . 5 ⊢ (1 ∈ ℝ → (1 + (0 −ℝ 1)) = 0) | |
14 | 8, 13 | ax-mp 5 | . . . 4 ⊢ (1 + (0 −ℝ 1)) = 0 |
15 | 14 | oveq2i 7202 | . . 3 ⊢ (i · (1 + (0 −ℝ 1))) = (i · 0) |
16 | sn-it0e0 40046 | . . 3 ⊢ (i · 0) = 0 | |
17 | 15, 16 | eqtri 2759 | . 2 ⊢ (i · (1 + (0 −ℝ 1))) = 0 |
18 | 5, 12, 17 | 3eqtr2i 2765 | 1 ⊢ (i + (i · (i · i))) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 (class class class)co 7191 ℝcr 10693 0cc0 10694 1c1 10695 ici 10696 + caddc 10697 · cmul 10699 −ℝ cresub 39997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-po 5453 df-so 5454 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-ltxr 10837 df-2 11858 df-3 11859 df-resub 39998 |
This theorem is referenced by: sn-0tie0 40070 sn-inelr 40084 |
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