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Mirrors > Home > MPE Home > Th. List > peano2cnm | Structured version Visualization version GIF version |
Description: "Reverse" second Peano postulate analogue for complex numbers: A complex number minus 1 is a complex number. (Contributed by Alexander van der Vekens, 18-Mar-2018.) |
Ref | Expression |
---|---|
peano2cnm | ⊢ (𝑁 ∈ ℂ → (𝑁 − 1) ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 10595 | . 2 ⊢ 1 ∈ ℂ | |
2 | subcl 10885 | . 2 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑁 − 1) ∈ ℂ) | |
3 | 1, 2 | mpan2 689 | 1 ⊢ (𝑁 ∈ ℂ → (𝑁 − 1) ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 (class class class)co 7156 ℂcc 10535 1c1 10538 − cmin 10870 |
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 |
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-ltxr 10680 df-sub 10872 |
This theorem is referenced by: kcnktkm1cn 11071 xp1d2m1eqxm1d2 11892 elnnnn0 11941 hash2iun1dif1 15179 pwdif 15223 pwm1geoserOLD 15225 nn0ob 15735 2lgslem1a1 25965 addsqrexnreu 26018 addsqnreup 26019 addsq2nreurex 26020 clwlkclwwlklem2a1 27770 clwlkclwwlklem2a 27776 clwlkclwwlklem3 27779 frrusgrord0 28119 numclwwlk7 28170 dirkertrigeqlem2 42433 fmtnoprmfac2 43778 lighneallem3 43821 proththd 43828 zofldiv2ALTV 43876 nn0onn0exALTV 43913 nn0onn0ex 44632 nn0sumshdiglemB 44729 |
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