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Mirrors > Home > MPE Home > Th. List > zq | Structured version Visualization version GIF version |
Description: An integer is a rational number. (Contributed by NM, 9-Jan-2002.) (Proof shortened by Steven Nguyen, 23-Mar-2023.) |
Ref | Expression |
---|---|
zq | ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 12067 | . . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
2 | 1 | div1d 11486 | . 2 ⊢ (𝐴 ∈ ℤ → (𝐴 / 1) = 𝐴) |
3 | 1nn 11727 | . . 3 ⊢ 1 ∈ ℕ | |
4 | znq 12434 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 1 ∈ ℕ) → (𝐴 / 1) ∈ ℚ) | |
5 | 3, 4 | mpan2 691 | . 2 ⊢ (𝐴 ∈ ℤ → (𝐴 / 1) ∈ ℚ) |
6 | 2, 5 | eqeltrrd 2834 | 1 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 (class class class)co 7170 1c1 10616 / cdiv 11375 ℕcn 11716 ℤcz 12062 ℚcq 12430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-z 12063 df-q 12431 |
This theorem is referenced by: zssq 12438 qbtwnxr 12676 modirr 13401 qexpcl 13537 qexpclz 13542 zsqrtelqelz 16198 pczpre 16284 pc0 16291 pcrec 16295 pcdvdstr 16312 pcgcd1 16313 pcgcd 16314 pc2dvds 16315 pc11 16316 sylow1lem1 18841 vitalilem1 24360 elqaalem1 25067 elqaalem3 25069 qaa 25071 2irrexpq 25473 2logb9irrALT 25536 2irrexpqALT 25538 lgsneg 26057 lgsdilem2 26069 lgsne0 26071 2sq2 26169 qabvle 26361 ostthlem1 26363 ostthlem2 26364 padicabv 26366 ostth2lem2 26370 ostth2 26373 ostth3 26374 qqhucn 31512 irrdifflemf 35116 irrdiff 35117 mblfinlem1 35437 oexpreposd 39897 zrtelqelz 39922 rmxypairf1o 40305 rmxycomplete 40311 rmxyadd 40315 rmxy1 40316 mpaaeu 40547 aacllem 45958 |
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