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Mirrors > Home > MPE Home > Th. List > eirr | Structured version Visualization version GIF version |
Description: e is irrational. (Contributed by Paul Chapman, 9-Feb-2008.) (Proof shortened by Mario Carneiro, 29-Apr-2014.) |
Ref | Expression |
---|---|
eirr | ⊢ e ∉ ℚ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛))) | |
2 | simpll 766 | . . . . . . 7 ⊢ (((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) ∧ e = (𝑝 / 𝑞)) → 𝑝 ∈ ℤ) | |
3 | simplr 768 | . . . . . . 7 ⊢ (((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) ∧ e = (𝑝 / 𝑞)) → 𝑞 ∈ ℕ) | |
4 | simpr 484 | . . . . . . 7 ⊢ (((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) ∧ e = (𝑝 / 𝑞)) → e = (𝑝 / 𝑞)) | |
5 | 1, 2, 3, 4 | eirrlem 16174 | . . . . . 6 ⊢ ¬ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) ∧ e = (𝑝 / 𝑞)) |
6 | 5 | imnani 400 | . . . . 5 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → ¬ e = (𝑝 / 𝑞)) |
7 | 6 | nrexdv 3145 | . . . 4 ⊢ (𝑝 ∈ ℤ → ¬ ∃𝑞 ∈ ℕ e = (𝑝 / 𝑞)) |
8 | 7 | nrex 3070 | . . 3 ⊢ ¬ ∃𝑝 ∈ ℤ ∃𝑞 ∈ ℕ e = (𝑝 / 𝑞) |
9 | elq 12958 | . . 3 ⊢ (e ∈ ℚ ↔ ∃𝑝 ∈ ℤ ∃𝑞 ∈ ℕ e = (𝑝 / 𝑞)) | |
10 | 8, 9 | mtbir 323 | . 2 ⊢ ¬ e ∈ ℚ |
11 | 10 | nelir 3045 | 1 ⊢ e ∉ ℚ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∉ wnel 3042 ∃wrex 3066 ↦ cmpt 5225 ‘cfv 6542 (class class class)co 7414 1c1 11133 / cdiv 11895 ℕcn 12236 ℕ0cn0 12496 ℤcz 12582 ℚcq 12956 !cfa 14258 eceu 16032 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9658 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-inf 9460 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-n0 12497 df-z 12583 df-uz 12847 df-q 12957 df-rp 13001 df-ico 13356 df-fz 13511 df-fzo 13654 df-fl 13783 df-seq 13993 df-exp 14053 df-fac 14259 df-bc 14288 df-hash 14316 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15441 df-clim 15458 df-rlim 15459 df-sum 15659 df-ef 16037 df-e 16038 |
This theorem is referenced by: egt2lt3 16176 |
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