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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 2824 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛))) | |
2 | simpll 765 | . . . . . . 7 ⊢ (((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) ∧ e = (𝑝 / 𝑞)) → 𝑝 ∈ ℤ) | |
3 | simplr 767 | . . . . . . 7 ⊢ (((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) ∧ e = (𝑝 / 𝑞)) → 𝑞 ∈ ℕ) | |
4 | simpr 487 | . . . . . . 7 ⊢ (((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) ∧ e = (𝑝 / 𝑞)) → e = (𝑝 / 𝑞)) | |
5 | 1, 2, 3, 4 | eirrlem 15560 | . . . . . 6 ⊢ ¬ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) ∧ e = (𝑝 / 𝑞)) |
6 | 5 | imnani 403 | . . . . 5 ⊢ ((𝑝 ∈ ℤ ∧ 𝑞 ∈ ℕ) → ¬ e = (𝑝 / 𝑞)) |
7 | 6 | nrexdv 3273 | . . . 4 ⊢ (𝑝 ∈ ℤ → ¬ ∃𝑞 ∈ ℕ e = (𝑝 / 𝑞)) |
8 | 7 | nrex 3272 | . . 3 ⊢ ¬ ∃𝑝 ∈ ℤ ∃𝑞 ∈ ℕ e = (𝑝 / 𝑞) |
9 | elq 12353 | . . 3 ⊢ (e ∈ ℚ ↔ ∃𝑝 ∈ ℤ ∃𝑞 ∈ ℕ e = (𝑝 / 𝑞)) | |
10 | 8, 9 | mtbir 325 | . 2 ⊢ ¬ e ∈ ℚ |
11 | 10 | nelir 3129 | 1 ⊢ e ∉ ℚ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∉ wnel 3126 ∃wrex 3142 ↦ cmpt 5149 ‘cfv 6358 (class class class)co 7159 1c1 10541 / cdiv 11300 ℕcn 11641 ℕ0cn0 11900 ℤcz 11984 ℚcq 12351 !cfa 13636 eceu 15419 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-pm 8412 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-ico 12747 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14429 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-limsup 14831 df-clim 14848 df-rlim 14849 df-sum 15046 df-ef 15424 df-e 15425 |
This theorem is referenced by: egt2lt3 15562 |
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