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Mirrors > Home > MPE Home > Th. List > Mathboxes > reelznn0nn | Structured version Visualization version GIF version |
Description: elznn0nn 12471 restated using df-resub 40737. (Contributed by SN, 25-Jan-2025.) |
Ref | Expression |
---|---|
reelznn0nn | ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ (0 −ℝ 𝑁) ∈ ℕ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elznn0nn 12471 | . 2 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))) | |
2 | df-neg 11346 | . . . . . 6 ⊢ -𝑁 = (0 − 𝑁) | |
3 | 0re 11115 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
4 | resubeqsub 40800 | . . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 −ℝ 𝑁) = (0 − 𝑁)) | |
5 | 3, 4 | mpan 688 | . . . . . 6 ⊢ (𝑁 ∈ ℝ → (0 −ℝ 𝑁) = (0 − 𝑁)) |
6 | 2, 5 | eqtr4id 2796 | . . . . 5 ⊢ (𝑁 ∈ ℝ → -𝑁 = (0 −ℝ 𝑁)) |
7 | 6 | eleq1d 2822 | . . . 4 ⊢ (𝑁 ∈ ℝ → (-𝑁 ∈ ℕ ↔ (0 −ℝ 𝑁) ∈ ℕ)) |
8 | 7 | pm5.32i 575 | . . 3 ⊢ ((𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) ↔ (𝑁 ∈ ℝ ∧ (0 −ℝ 𝑁) ∈ ℕ)) |
9 | 8 | orbi2i 911 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)) ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ (0 −ℝ 𝑁) ∈ ℕ))) |
10 | 1, 9 | bitri 274 | 1 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ0 ∨ (𝑁 ∈ ℝ ∧ (0 −ℝ 𝑁) ∈ ℕ))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 (class class class)co 7351 ℝcr 11008 0cc0 11009 − cmin 11343 -cneg 11344 ℕcn 12111 ℕ0cn0 12371 ℤcz 12457 −ℝ cresub 40736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-ltxr 11152 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-n0 12372 df-z 12458 df-resub 40737 |
This theorem is referenced by: zaddcom 40823 zmulcom 40827 |
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