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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 | zcn 9027 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
2 | 1 | div1d 8508 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥 / 1) = 𝑥) |
3 | 2 | eqeq2d 2129 | . . . . 5 ⊢ (𝑥 ∈ ℤ → (𝐴 = (𝑥 / 1) ↔ 𝐴 = 𝑥)) |
4 | eqcom 2119 | . . . . 5 ⊢ (𝑥 = 𝐴 ↔ 𝐴 = 𝑥) | |
5 | 3, 4 | syl6rbbr 198 | . . . 4 ⊢ (𝑥 ∈ ℤ → (𝑥 = 𝐴 ↔ 𝐴 = (𝑥 / 1))) |
6 | 1nn 8699 | . . . . 5 ⊢ 1 ∈ ℕ | |
7 | oveq2 5750 | . . . . . . 7 ⊢ (𝑦 = 1 → (𝑥 / 𝑦) = (𝑥 / 1)) | |
8 | 7 | eqeq2d 2129 | . . . . . 6 ⊢ (𝑦 = 1 → (𝐴 = (𝑥 / 𝑦) ↔ 𝐴 = (𝑥 / 1))) |
9 | 8 | rspcev 2763 | . . . . 5 ⊢ ((1 ∈ ℕ ∧ 𝐴 = (𝑥 / 1)) → ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
10 | 6, 9 | mpan 420 | . . . 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 2504 | . 2 ⊢ (∃𝑥 ∈ ℤ 𝑥 = 𝐴 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
13 | risset 2440 | . 2 ⊢ (𝐴 ∈ ℤ ↔ ∃𝑥 ∈ ℤ 𝑥 = 𝐴) | |
14 | elq 9382 | . 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 1316 ∈ wcel 1465 ∃wrex 2394 (class class class)co 5742 1c1 7589 / cdiv 8400 ℕcn 8688 ℤcz 9022 ℚcq 9379 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-z 9023 df-q 9380 |
This theorem is referenced by: zssq 9387 qdivcl 9403 irrmul 9407 qbtwnz 9997 qbtwnxr 10003 flqlt 10024 flid 10025 flqltnz 10028 flqbi2 10032 flqaddz 10038 flqmulnn0 10040 ceilid 10056 flqeqceilz 10059 flqdiv 10062 modqcl 10067 mulqmod0 10071 modqfrac 10078 zmod10 10081 modqmulnn 10083 zmodcl 10085 zmodfz 10087 zmodid2 10093 q0mod 10096 q1mod 10097 modqcyc 10100 mulp1mod1 10106 modqmuladd 10107 modqmuladdim 10108 modqmuladdnn0 10109 m1modnnsub1 10111 addmodid 10113 modqm1p1mod0 10116 modqltm1p1mod 10117 modqmul1 10118 modqmul12d 10119 q2txmodxeq0 10125 modifeq2int 10127 modaddmodup 10128 modaddmodlo 10129 modqaddmulmod 10132 modqdi 10133 modqsubdir 10134 modsumfzodifsn 10137 addmodlteq 10139 qexpcl 10277 qexpclz 10282 iexpcyc 10365 facavg 10460 bcval 10463 qabsor 10815 modfsummodlemstep 11194 egt2lt3 11413 dvdsval3 11424 moddvds 11429 absdvdsb 11438 dvdsabsb 11439 dvdslelemd 11468 dvdsmod 11487 mulmoddvds 11488 divalglemnn 11542 divalgmod 11551 fldivndvdslt 11559 gcdabs 11603 gcdabs1 11604 modgcd 11606 bezoutlemnewy 11611 bezoutlemstep 11612 eucalglt 11665 lcmabs 11684 sqrt2irraplemnn 11784 nn0sqrtelqelz 11811 crth 11827 phimullem 11828 |
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