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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 8356 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
2 | 1 | div1d 7868 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → (𝑥 / 1) = 𝑥) |
3 | 2 | eqeq2d 2092 | . . . . 5 ⊢ (𝑥 ∈ ℤ → (𝐴 = (𝑥 / 1) ↔ 𝐴 = 𝑥)) |
4 | eqcom 2083 | . . . . 5 ⊢ (𝑥 = 𝐴 ↔ 𝐴 = 𝑥) | |
5 | 3, 4 | syl6rbbr 197 | . . . 4 ⊢ (𝑥 ∈ ℤ → (𝑥 = 𝐴 ↔ 𝐴 = (𝑥 / 1))) |
6 | 1nn 8050 | . . . . 5 ⊢ 1 ∈ ℕ | |
7 | oveq2 5540 | . . . . . . 7 ⊢ (𝑦 = 1 → (𝑥 / 𝑦) = (𝑥 / 1)) | |
8 | 7 | eqeq2d 2092 | . . . . . 6 ⊢ (𝑦 = 1 → (𝐴 = (𝑥 / 𝑦) ↔ 𝐴 = (𝑥 / 1))) |
9 | 8 | rspcev 2701 | . . . . 5 ⊢ ((1 ∈ ℕ ∧ 𝐴 = (𝑥 / 1)) → ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
10 | 6, 9 | mpan 414 | . . . 4 ⊢ (𝐴 = (𝑥 / 1) → ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
11 | 5, 10 | syl6bi 161 | . . 3 ⊢ (𝑥 ∈ ℤ → (𝑥 = 𝐴 → ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦))) |
12 | 11 | reximia 2456 | . 2 ⊢ (∃𝑥 ∈ ℤ 𝑥 = 𝐴 → ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) |
13 | risset 2394 | . 2 ⊢ (𝐴 ∈ ℤ ↔ ∃𝑥 ∈ ℤ 𝑥 = 𝐴) | |
14 | elq 8707 | . 2 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) | |
15 | 12, 13, 14 | 3imtr4i 199 | 1 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∈ wcel 1433 ∃wrex 2349 (class class class)co 5532 1c1 6982 / cdiv 7760 ℕcn 8039 ℤcz 8351 ℚcq 8704 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-z 8352 df-q 8705 |
This theorem is referenced by: zssq 8712 qdivcl 8728 irrmul 8732 qbtwnzlemstep 9257 qbtwnxr 9266 flqlt 9285 flid 9286 flqltnz 9289 flqbi2 9293 flqaddz 9299 flqmulnn0 9301 ceilid 9317 flqeqceilz 9320 flqdiv 9323 modqcl 9328 mulqmod0 9332 modqfrac 9339 zmod10 9342 modqmulnn 9344 zmodcl 9346 zmodfz 9348 zmodid2 9354 q0mod 9357 q1mod 9358 modqcyc 9361 mulp1mod1 9367 modqmuladd 9368 modqmuladdim 9369 modqmuladdnn0 9370 m1modnnsub1 9372 addmodid 9374 modqm1p1mod0 9377 modqltm1p1mod 9378 modqmul1 9379 modqmul12d 9380 q2txmodxeq0 9386 modifeq2int 9388 modaddmodup 9389 modaddmodlo 9390 modqaddmulmod 9393 modqdi 9394 modqsubdir 9395 modsumfzodifsn 9398 addmodlteq 9400 qexpcl 9492 qexpclz 9497 iexpcyc 9579 facavg 9673 bcval 9676 qabsor 9961 dvdsval3 10199 moddvds 10204 absdvdsb 10213 dvdsabsb 10214 dvdslelemd 10243 dvdsmod 10262 mulmoddvds 10263 divalglemnn 10318 divalgmod 10327 fldivndvdslt 10335 gcdabs 10379 gcdabs1 10380 modgcd 10382 bezoutlemnewy 10385 bezoutlemstep 10386 eucalglt 10439 lcmabs 10458 sqrt2irraplemnn 10557 |
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