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Mirrors > Home > MPE Home > Th. List > Mathboxes > reixi | Structured version Visualization version GIF version |
Description: ixi 11849 without ax-mulcom 11178. (Contributed by SN, 5-May-2024.) |
Ref | Expression |
---|---|
reixi | ⊢ (i · i) = (0 −ℝ 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-i2m1 11182 | . . . 4 ⊢ ((i · i) + 1) = 0 | |
2 | 1re 11220 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | renegid2 41590 | . . . . 5 ⊢ (1 ∈ ℝ → ((0 −ℝ 1) + 1) = 0) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ((0 −ℝ 1) + 1) = 0 |
5 | 1, 4 | eqtr4i 2761 | . . 3 ⊢ ((i · i) + 1) = ((0 −ℝ 1) + 1) |
6 | ax-icn 11173 | . . . . . 6 ⊢ i ∈ ℂ | |
7 | 6, 6 | mulcli 11227 | . . . . 5 ⊢ (i · i) ∈ ℂ |
8 | 7 | a1i 11 | . . . 4 ⊢ (⊤ → (i · i) ∈ ℂ) |
9 | rernegcl 41548 | . . . . . 6 ⊢ (1 ∈ ℝ → (0 −ℝ 1) ∈ ℝ) | |
10 | 9 | recnd 11248 | . . . . 5 ⊢ (1 ∈ ℝ → (0 −ℝ 1) ∈ ℂ) |
11 | 2, 10 | mp1i 13 | . . . 4 ⊢ (⊤ → (0 −ℝ 1) ∈ ℂ) |
12 | 1cnd 11215 | . . . 4 ⊢ (⊤ → 1 ∈ ℂ) | |
13 | 8, 11, 12 | sn-addcan2d 41598 | . . 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 1546 | 1 ⊢ (i · i) = (0 −ℝ 1) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ⊤wtru 1540 ∈ wcel 2104 (class class class)co 7413 ℂcc 11112 ℝcr 11113 0cc0 11114 1c1 11115 ici 11116 + caddc 11117 · cmul 11119 −ℝ cresub 41542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11256 df-mnf 11257 df-ltxr 11259 df-2 12281 df-3 12282 df-resub 41543 |
This theorem is referenced by: rei4 41600 ipiiie0 41614 sn-0tie0 41616 sn-inelr 41642 |
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