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Mathbox for Thierry Arnoux |
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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 12415 | . . . . 5 ⊢ 0 ≠ 2 | |
2 | 1 | nesymi 2990 | . . . 4 ⊢ ¬ 2 = 0 |
3 | oveq2 7409 | . . . . 5 ⊢ (1 = -1 → (1 + 1) = (1 + -1)) | |
4 | 1p1e2 12333 | . . . . 5 ⊢ (1 + 1) = 2 | |
5 | 1pneg1e0 12327 | . . . . 5 ⊢ (1 + -1) = 0 | |
6 | 3, 4, 5 | 3eqtr3g 2787 | . . . 4 ⊢ (1 = -1 → 2 = 0) |
7 | 2, 6 | mto 196 | . . 3 ⊢ ¬ 1 = -1 |
8 | id 22 | . . . . 5 ⊢ (1 = i → 1 = i) | |
9 | 8, 8 | oveq12d 7419 | . . . 4 ⊢ (1 = i → (1 · 1) = (i · i)) |
10 | 1t1e1 12370 | . . . 4 ⊢ (1 · 1) = 1 | |
11 | ixi 11839 | . . . 4 ⊢ (i · i) = -1 | |
12 | 9, 10, 11 | 3eqtr3g 2787 | . . 3 ⊢ (1 = i → 1 = -1) |
13 | 7, 12 | mto 196 | . 2 ⊢ ¬ 1 = i |
14 | 13 | neir 2935 | 1 ⊢ 1 ≠ i |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ≠ wne 2932 (class class class)co 7401 0cc0 11105 1c1 11106 ici 11107 + caddc 11108 · cmul 11110 -cneg 11441 2c2 12263 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 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 ax-pre-mulgt0 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-2 12271 |
This theorem is referenced by: ccfldextdgrr 33192 |
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