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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 11978 | . . 3 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℂ) | |
2 | 1 | div1d 11400 | . 2 ⊢ (𝐴 ∈ ℤ → (𝐴 / 1) = 𝐴) |
3 | 1nn 11641 | . . 3 ⊢ 1 ∈ ℕ | |
4 | znq 12344 | . . 3 ⊢ ((𝐴 ∈ ℤ ∧ 1 ∈ ℕ) → (𝐴 / 1) ∈ ℚ) | |
5 | 3, 4 | mpan2 689 | . 2 ⊢ (𝐴 ∈ ℤ → (𝐴 / 1) ∈ ℚ) |
6 | 2, 5 | eqeltrrd 2912 | 1 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2108 (class class class)co 7148 1c1 10530 / cdiv 11289 ℕcn 11630 ℤcz 11973 ℚcq 12340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-z 11974 df-q 12341 |
This theorem is referenced by: zssq 12347 qbtwnxr 12585 modirr 13302 qexpcl 13437 qexpclz 13442 zsqrtelqelz 16090 pczpre 16176 pc0 16183 pcrec 16187 pcdvdstr 16204 pcgcd1 16205 pcgcd 16206 pc2dvds 16207 pc11 16208 sylow1lem1 18715 vitalilem1 24201 elqaalem1 24900 elqaalem3 24902 qaa 24904 2irrexpq 25305 2logb9irrALT 25368 2irrexpqALT 25370 lgsneg 25889 lgsdilem2 25901 lgsne0 25903 2sq2 26001 qabvle 26193 ostthlem1 26195 ostthlem2 26196 padicabv 26198 ostth2lem2 26202 ostth2 26205 ostth3 26206 qqhucn 31226 mblfinlem1 34921 oexpreposd 39170 zrtelqelz 39183 rmxypairf1o 39499 rmxycomplete 39505 rmxyadd 39509 rmxy1 39510 mpaaeu 39741 aacllem 44893 |
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