Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > qre | GIF version |
Description: A rational number is a real number. (Contributed by NM, 14-Nov-2002.) |
Ref | Expression |
---|---|
qre | ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elq 8707 | . 2 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) | |
2 | zre 8355 | . . . . 5 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ) | |
3 | nnre 8046 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ) | |
4 | nnap0 8068 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 # 0) | |
5 | 3, 4 | jca 300 | . . . . 5 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ ℝ ∧ 𝑦 # 0)) |
6 | redivclap 7819 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑦 # 0) → (𝑥 / 𝑦) ∈ ℝ) | |
7 | 6 | 3expb 1139 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ (𝑦 ∈ ℝ ∧ 𝑦 # 0)) → (𝑥 / 𝑦) ∈ ℝ) |
8 | 2, 5, 7 | syl2an 283 | . . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑥 / 𝑦) ∈ ℝ) |
9 | eleq1 2141 | . . . 4 ⊢ (𝐴 = (𝑥 / 𝑦) → (𝐴 ∈ ℝ ↔ (𝑥 / 𝑦) ∈ ℝ)) | |
10 | 8, 9 | syl5ibrcom 155 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝐴 = (𝑥 / 𝑦) → 𝐴 ∈ ℝ)) |
11 | 10 | rexlimivv 2482 | . 2 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) → 𝐴 ∈ ℝ) |
12 | 1, 11 | sylbi 119 | 1 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 ∃wrex 2349 class class class wbr 3785 (class class class)co 5532 ℝcr 6980 0cc0 6981 # cap 7681 / 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: qssre 8715 qltlen 8725 qlttri2 8726 irradd 8731 irrmul 8732 qletric 9253 qlelttric 9254 qltnle 9255 qdceq 9256 qbtwnzlemstep 9257 qbtwnzlemex 9259 qbtwnz 9260 qbtwnxr 9266 qavgle 9267 ioo0 9268 ioom 9269 ico0 9270 ioc0 9271 flqcl 9277 flqlelt 9278 qfraclt1 9282 qfracge0 9283 flqge 9284 flqltnz 9289 flqwordi 9290 flqbi 9292 flqbi2 9293 flqaddz 9299 flqmulnn0 9301 flltdivnn0lt 9306 ceilqval 9308 ceiqge 9311 ceiqm1l 9313 ceiqle 9315 flqleceil 9319 flqeqceilz 9320 intfracq 9322 flqdiv 9323 modqval 9326 modq0 9331 mulqmod0 9332 negqmod0 9333 modqge0 9334 modqlt 9335 modqelico 9336 modqdiffl 9337 modqmulnn 9344 modqid 9351 modqid0 9352 modqabs 9359 modqabs2 9360 modqcyc 9361 mulqaddmodid 9366 modqmuladdim 9369 modqmuladdnn0 9370 modqltm1p1mod 9378 q2txmodxeq0 9386 q2submod 9387 modqdi 9394 modqsubdir 9395 qabsor 9961 qdenre 10088 flodddiv4t2lthalf 10337 sqrt2irraplemnn 10557 sqrt2irrap 10558 qdencn 10785 |
Copyright terms: Public domain | W3C validator |