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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 8956 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
2 | 1 | div1d 8446 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥 / 1) = 𝑥) |
3 | 2 | eqeq2d 2124 | . . . . 5 ⊢ (𝑥 ∈ ℤ → (𝐴 = (𝑥 / 1) ↔ 𝐴 = 𝑥)) |
4 | eqcom 2115 | . . . . 5 ⊢ (𝑥 = 𝐴 ↔ 𝐴 = 𝑥) | |
5 | 3, 4 | syl6rbbr 198 | . . . 4 ⊢ (𝑥 ∈ ℤ → (𝑥 = 𝐴 ↔ 𝐴 = (𝑥 / 1))) |
6 | 1nn 8634 | . . . . 5 ⊢ 1 ∈ ℕ | |
7 | oveq2 5734 | . . . . . . 7 ⊢ (𝑦 = 1 → (𝑥 / 𝑦) = (𝑥 / 1)) | |
8 | 7 | eqeq2d 2124 | . . . . . 6 ⊢ (𝑦 = 1 → (𝐴 = (𝑥 / 𝑦) ↔ 𝐴 = (𝑥 / 1))) |
9 | 8 | rspcev 2758 | . . . . 5 ⊢ ((1 ∈ ℕ ∧ 𝐴 = (𝑥 / 1)) → ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
10 | 6, 9 | mpan 418 | . . . 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 2499 | . 2 ⊢ (∃𝑥 ∈ ℤ 𝑥 = 𝐴 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
13 | risset 2435 | . 2 ⊢ (𝐴 ∈ ℤ ↔ ∃𝑥 ∈ ℤ 𝑥 = 𝐴) | |
14 | elq 9309 | . 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 1312 ∈ wcel 1461 ∃wrex 2389 (class class class)co 5726 1c1 7541 / cdiv 8338 ℕcn 8623 ℤcz 8951 ℚcq 9306 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7629 ax-resscn 7630 ax-1cn 7631 ax-1re 7632 ax-icn 7633 ax-addcl 7634 ax-addrcl 7635 ax-mulcl 7636 ax-mulrcl 7637 ax-addcom 7638 ax-mulcom 7639 ax-addass 7640 ax-mulass 7641 ax-distr 7642 ax-i2m1 7643 ax-0lt1 7644 ax-1rid 7645 ax-0id 7646 ax-rnegex 7647 ax-precex 7648 ax-cnre 7649 ax-pre-ltirr 7650 ax-pre-ltwlin 7651 ax-pre-lttrn 7652 ax-pre-apti 7653 ax-pre-ltadd 7654 ax-pre-mulgt0 7655 ax-pre-mulext 7656 |
This theorem depends on definitions: df-bi 116 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rmo 2396 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-id 4173 df-po 4176 df-iso 4177 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5989 df-2nd 5990 df-pnf 7719 df-mnf 7720 df-xr 7721 df-ltxr 7722 df-le 7723 df-sub 7851 df-neg 7852 df-reap 8248 df-ap 8255 df-div 8339 df-inn 8624 df-z 8952 df-q 9307 |
This theorem is referenced by: zssq 9314 qdivcl 9330 irrmul 9334 qbtwnz 9915 qbtwnxr 9921 flqlt 9942 flid 9943 flqltnz 9946 flqbi2 9950 flqaddz 9956 flqmulnn0 9958 ceilid 9974 flqeqceilz 9977 flqdiv 9980 modqcl 9985 mulqmod0 9989 modqfrac 9996 zmod10 9999 modqmulnn 10001 zmodcl 10003 zmodfz 10005 zmodid2 10011 q0mod 10014 q1mod 10015 modqcyc 10018 mulp1mod1 10024 modqmuladd 10025 modqmuladdim 10026 modqmuladdnn0 10027 m1modnnsub1 10029 addmodid 10031 modqm1p1mod0 10034 modqltm1p1mod 10035 modqmul1 10036 modqmul12d 10037 q2txmodxeq0 10043 modifeq2int 10045 modaddmodup 10046 modaddmodlo 10047 modqaddmulmod 10050 modqdi 10051 modqsubdir 10052 modsumfzodifsn 10055 addmodlteq 10057 qexpcl 10195 qexpclz 10200 iexpcyc 10283 facavg 10378 bcval 10381 qabsor 10732 modfsummodlemstep 11111 egt2lt3 11327 dvdsval3 11338 moddvds 11343 absdvdsb 11352 dvdsabsb 11353 dvdslelemd 11382 dvdsmod 11401 mulmoddvds 11402 divalglemnn 11456 divalgmod 11465 fldivndvdslt 11473 gcdabs 11517 gcdabs1 11518 modgcd 11520 bezoutlemnewy 11523 bezoutlemstep 11524 eucalglt 11577 lcmabs 11596 sqrt2irraplemnn 11695 nn0sqrtelqelz 11722 crth 11738 phimullem 11739 |
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