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| Mirrors > Home > MPE Home > Th. List > qre | Structured version Visualization version 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 12992 | . 2 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) | |
| 2 | zre 12617 | . . . . 5 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℝ) | |
| 3 | nnre 12273 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ) | |
| 4 | nnne0 12300 | . . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0) | |
| 5 | 3, 4 | jca 511 | . . . . 5 ⊢ (𝑦 ∈ ℕ → (𝑦 ∈ ℝ ∧ 𝑦 ≠ 0)) |
| 6 | redivcl 11986 | . . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑦 ≠ 0) → (𝑥 / 𝑦) ∈ ℝ) | |
| 7 | 6 | 3expb 1121 | . . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ (𝑦 ∈ ℝ ∧ 𝑦 ≠ 0)) → (𝑥 / 𝑦) ∈ ℝ) |
| 8 | 2, 5, 7 | syl2an 596 | . . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝑥 / 𝑦) ∈ ℝ) |
| 9 | eleq1 2829 | . . . 4 ⊢ (𝐴 = (𝑥 / 𝑦) → (𝐴 ∈ ℝ ↔ (𝑥 / 𝑦) ∈ ℝ)) | |
| 10 | 8, 9 | syl5ibrcom 247 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝐴 = (𝑥 / 𝑦) → 𝐴 ∈ ℝ)) |
| 11 | 10 | rexlimivv 3201 | . 2 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) → 𝐴 ∈ ℝ) |
| 12 | 1, 11 | sylbi 217 | 1 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 (class class class)co 7431 ℝcr 11154 0cc0 11155 / cdiv 11920 ℕcn 12266 ℤcz 12613 ℚcq 12990 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-z 12614 df-q 12991 |
| This theorem is referenced by: qred 12997 qssre 13001 irradd 13015 irrmul 13016 qbtwnxr 13242 qsqueeze 13243 qextltlem 13244 xralrple 13247 ixxub 13408 ixxlb 13409 ioo0 13412 ico0 13433 ioc0 13434 qnumgt0 16787 pcabs 16913 blssps 24434 blss 24435 blcld 24518 qdensere 24790 nmoleub2lem3 25148 mbfaddlem 25695 dvlip2 26034 itgsubst 26090 aalioulem2 26375 aalioulem4 26377 aalioulem5 26378 aalioulem6 26379 aaliou 26380 aaliou2b 26383 ipasslem8 30856 2sqr3minply 33791 irrdifflemf 37326 itg2gt0cn 37682 3cubeslem1 42695 3cubeslem2 42696 3cubeslem3r 42698 3cubeslem4 42700 irrapxlem5 42837 rpnnen3lem 43043 qinioo 45548 qelioo 45559 qndenserrnbllem 46309 smfaddlem1 46778 smfaddlem2 46779 |
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