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Mirrors > Home > MPE Home > Th. List > Mathboxes > reixi | Structured version Visualization version GIF version |
Description: ixi 11838 without ax-mulcom 11169. (Contributed by SN, 5-May-2024.) |
Ref | Expression |
---|---|
reixi | ⊢ (i · i) = (0 −ℝ 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-i2m1 11173 | . . . 4 ⊢ ((i · i) + 1) = 0 | |
2 | 1re 11209 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | renegid2 41229 | . . . . 5 ⊢ (1 ∈ ℝ → ((0 −ℝ 1) + 1) = 0) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ((0 −ℝ 1) + 1) = 0 |
5 | 1, 4 | eqtr4i 2764 | . . 3 ⊢ ((i · i) + 1) = ((0 −ℝ 1) + 1) |
6 | ax-icn 11164 | . . . . . 6 ⊢ i ∈ ℂ | |
7 | 6, 6 | mulcli 11216 | . . . . 5 ⊢ (i · i) ∈ ℂ |
8 | 7 | a1i 11 | . . . 4 ⊢ (⊤ → (i · i) ∈ ℂ) |
9 | rernegcl 41187 | . . . . . 6 ⊢ (1 ∈ ℝ → (0 −ℝ 1) ∈ ℝ) | |
10 | 9 | recnd 11237 | . . . . 5 ⊢ (1 ∈ ℝ → (0 −ℝ 1) ∈ ℂ) |
11 | 2, 10 | mp1i 13 | . . . 4 ⊢ (⊤ → (0 −ℝ 1) ∈ ℂ) |
12 | 1cnd 11204 | . . . 4 ⊢ (⊤ → 1 ∈ ℂ) | |
13 | 8, 11, 12 | sn-addcan2d 41237 | . . 3 ⊢ (⊤ → (((i · i) + 1) = ((0 −ℝ 1) + 1) ↔ (i · i) = (0 −ℝ 1))) |
14 | 5, 13 | mpbii 232 | . 2 ⊢ (⊤ → (i · i) = (0 −ℝ 1)) |
15 | 14 | mptru 1549 | 1 ⊢ (i · i) = (0 −ℝ 1) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 (class class class)co 7403 ℂcc 11103 ℝcr 11104 0cc0 11105 1c1 11106 ici 11107 + caddc 11108 · cmul 11110 −ℝ cresub 41181 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4907 df-br 5147 df-opab 5209 df-mpt 5230 df-id 5572 df-po 5586 df-so 5587 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11245 df-mnf 11246 df-ltxr 11248 df-2 12270 df-3 12271 df-resub 41182 |
This theorem is referenced by: rei4 41239 ipiiie0 41253 sn-0tie0 41255 sn-inelr 41281 |
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