| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > nnsscn | Structured version Visualization version GIF version | ||
| Description: The positive integers are a subset of the complex numbers. Remark: this could also be proven from nnssre 12270 and ax-resscn 11212 at the cost of using more axioms. (Contributed by NM, 2-Aug-2004.) Reduce dependencies on axioms. (Revised by Steven Nguyen, 4-Oct-2022.) |
| Ref | Expression |
|---|---|
| nnsscn | ⊢ ℕ ⊆ ℂ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ax-1cn 11213 | . 2 ⊢ 1 ∈ ℂ | |
| 2 | peano2cn 11433 | . . 3 ⊢ (𝑥 ∈ ℂ → (𝑥 + 1) ∈ ℂ) | |
| 3 | 2 | rgen 3063 | . 2 ⊢ ∀𝑥 ∈ ℂ (𝑥 + 1) ∈ ℂ |
| 4 | peano5nni 12269 | . 2 ⊢ ((1 ∈ ℂ ∧ ∀𝑥 ∈ ℂ (𝑥 + 1) ∈ ℂ) → ℕ ⊆ ℂ) | |
| 5 | 1, 3, 4 | mp2an 692 | 1 ⊢ ℕ ⊆ ℂ |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 ℕcn 12266 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-1cn 11213 ax-addcl 11215 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 |
| This theorem is referenced by: nnex 12272 nncn 12274 nncnd 12282 nn0sscn 12531 nn0addcl 12561 nn0mulcl 12562 dfz2 12632 nnexpcl 14115 fprodnncl 15991 nnrisefaccl 16055 znnen 16248 wunndx 17232 cmetcaulem 25322 mpodvdsmulf1o 27237 fsumdvdsmul 27238 dvdsmulf1o 27239 fsumdvdsmulOLD 27240 esumcvg 34087 eulerpartlemgs2 34382 fsum2dsub 34622 reprsuc 34630 nndivsub 36458 fsumnncl 45587 nnsgrpmgm 48092 nnsgrp 48093 nnsgrpnmnd 48094 |
| Copyright terms: Public domain | W3C validator |