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Mirrors > Home > ILE Home > Th. List > zq | GIF version |
Description: An integer is a rational number. (Contributed by NM, 9-Jan-2002.) |
Ref | Expression |
---|---|
zq | ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2166 | . . . . 5 ⊢ (𝑥 = 𝐴 ↔ 𝐴 = 𝑥) | |
2 | zcn 9187 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
3 | 2 | div1d 8667 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥 / 1) = 𝑥) |
4 | 3 | eqeq2d 2176 | . . . . 5 ⊢ (𝑥 ∈ ℤ → (𝐴 = (𝑥 / 1) ↔ 𝐴 = 𝑥)) |
5 | 1, 4 | bitr4id 198 | . . . 4 ⊢ (𝑥 ∈ ℤ → (𝑥 = 𝐴 ↔ 𝐴 = (𝑥 / 1))) |
6 | 1nn 8859 | . . . . 5 ⊢ 1 ∈ ℕ | |
7 | oveq2 5844 | . . . . . . 7 ⊢ (𝑦 = 1 → (𝑥 / 𝑦) = (𝑥 / 1)) | |
8 | 7 | eqeq2d 2176 | . . . . . 6 ⊢ (𝑦 = 1 → (𝐴 = (𝑥 / 𝑦) ↔ 𝐴 = (𝑥 / 1))) |
9 | 8 | rspcev 2825 | . . . . 5 ⊢ ((1 ∈ ℕ ∧ 𝐴 = (𝑥 / 1)) → ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
10 | 6, 9 | mpan 421 | . . . 4 ⊢ (𝐴 = (𝑥 / 1) → ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
11 | 5, 10 | syl6bi 162 | . . 3 ⊢ (𝑥 ∈ ℤ → (𝑥 = 𝐴 → ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))) |
12 | 11 | reximia 2559 | . 2 ⊢ (∃𝑥 ∈ ℤ 𝑥 = 𝐴 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
13 | risset 2492 | . 2 ⊢ (𝐴 ∈ ℤ ↔ ∃𝑥 ∈ ℤ 𝑥 = 𝐴) | |
14 | elq 9551 | . 2 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) | |
15 | 12, 13, 14 | 3imtr4i 200 | 1 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 ∃wrex 2443 (class class class)co 5836 1c1 7745 / cdiv 8559 ℕcn 8848 ℤcz 9182 ℚcq 9548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-z 9183 df-q 9549 |
This theorem is referenced by: zssq 9556 qdivcl 9572 irrmul 9576 qbtwnz 10177 qbtwnxr 10183 flqlt 10208 flid 10209 flqltnz 10212 flqbi2 10216 flqaddz 10222 flqmulnn0 10224 ceilid 10240 flqeqceilz 10243 flqdiv 10246 modqcl 10251 mulqmod0 10255 modqfrac 10262 zmod10 10265 modqmulnn 10267 zmodcl 10269 zmodfz 10271 zmodid2 10277 q0mod 10280 q1mod 10281 modqcyc 10284 mulp1mod1 10290 modqmuladd 10291 modqmuladdim 10292 modqmuladdnn0 10293 m1modnnsub1 10295 addmodid 10297 modqm1p1mod0 10300 modqltm1p1mod 10301 modqmul1 10302 modqmul12d 10303 q2txmodxeq0 10309 modifeq2int 10311 modaddmodup 10312 modaddmodlo 10313 modqaddmulmod 10316 modqdi 10317 modqsubdir 10318 modsumfzodifsn 10321 addmodlteq 10323 qexpcl 10461 qexpclz 10466 iexpcyc 10549 facavg 10648 bcval 10651 qabsor 11003 modfsummodlemstep 11384 egt2lt3 11706 dvdsval3 11717 p1modz1 11720 moddvds 11725 modm1div 11726 absdvdsb 11735 dvdsabsb 11736 dvdslelemd 11766 dvdsmod 11785 mulmoddvds 11786 divalglemnn 11840 divalgmod 11849 fldivndvdslt 11857 gcdabs 11906 gcdabs1 11907 modgcd 11909 bezoutlemnewy 11914 bezoutlemstep 11915 eucalglt 11968 lcmabs 11987 sqrt2irraplemnn 12090 nn0sqrtelqelz 12117 crth 12135 phimullem 12136 eulerthlema 12141 eulerthlemh 12142 fermltl 12145 prmdiv 12146 prmdiveq 12147 odzdvds 12156 vfermltl 12162 powm2modprm 12163 modprm0 12165 modprmn0modprm0 12167 pceu 12206 pczpre 12208 pcdiv 12213 pc0 12215 pcqdiv 12218 pcrec 12219 pcexp 12220 pcxcl 12222 pcdvdstr 12237 pcgcd1 12238 pc2dvds 12240 pc11 12241 pcaddlem 12249 pcadd 12250 fldivp1 12257 qexpz 12261 2logb9irrALT 13439 2irrexpq 13441 2irrexpqap 13443 apdifflemr 13767 apdiff 13768 |
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