Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > 1nei | Structured version Visualization version GIF version |
Description: The imaginary unit i is not one. (Contributed by Thierry Arnoux, 20-Aug-2023.) |
Ref | Expression |
---|---|
1nei | ⊢ 1 ≠ i |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ne2 11845 | . . . . 5 ⊢ 0 ≠ 2 | |
2 | 1 | nesymi 3073 | . . . 4 ⊢ ¬ 2 = 0 |
3 | oveq2 7164 | . . . . 5 ⊢ (1 = -1 → (1 + 1) = (1 + -1)) | |
4 | 1p1e2 11763 | . . . . 5 ⊢ (1 + 1) = 2 | |
5 | 1pneg1e0 11757 | . . . . 5 ⊢ (1 + -1) = 0 | |
6 | 3, 4, 5 | 3eqtr3g 2879 | . . . 4 ⊢ (1 = -1 → 2 = 0) |
7 | 2, 6 | mto 199 | . . 3 ⊢ ¬ 1 = -1 |
8 | id 22 | . . . . 5 ⊢ (1 = i → 1 = i) | |
9 | 8, 8 | oveq12d 7174 | . . . 4 ⊢ (1 = i → (1 · 1) = (i · i)) |
10 | 1t1e1 11800 | . . . 4 ⊢ (1 · 1) = 1 | |
11 | ixi 11269 | . . . 4 ⊢ (i · i) = -1 | |
12 | 9, 10, 11 | 3eqtr3g 2879 | . . 3 ⊢ (1 = i → 1 = -1) |
13 | 7, 12 | mto 199 | . 2 ⊢ ¬ 1 = i |
14 | 13 | neir 3019 | 1 ⊢ 1 ≠ i |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ≠ wne 3016 (class class class)co 7156 0cc0 10537 1c1 10538 ici 10539 + caddc 10540 · cmul 10542 -cneg 10871 2c2 11693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-2 11701 |
This theorem is referenced by: ccfldextdgrr 31057 |
Copyright terms: Public domain | W3C validator |