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Mirrors > Home > MPE Home > Th. List > Mathboxes > fltne | Structured version Visualization version GIF version |
Description: If a counterexample to FLT exists, its addends are not equal. (Contributed by Steven Nguyen, 1-Jun-2023.) |
Ref | Expression |
---|---|
fltne.a | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
fltne.b | ⊢ (𝜑 → 𝐵 ∈ ℕ) |
fltne.c | ⊢ (𝜑 → 𝐶 ∈ ℕ) |
fltne.n | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) |
fltne.1 | ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) |
Ref | Expression |
---|---|
fltne | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2prm 16036 | . . . . . 6 ⊢ 2 ∈ ℙ | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℙ) |
3 | fltne.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘3)) | |
4 | uzuzle23 12290 | . . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ (ℤ≥‘2)) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘2)) |
6 | rtprmirr 39243 | . . . . 5 ⊢ ((2 ∈ ℙ ∧ 𝑁 ∈ (ℤ≥‘2)) → (2↑𝑐(1 / 𝑁)) ∈ (ℝ ∖ ℚ)) | |
7 | 2, 5, 6 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (2↑𝑐(1 / 𝑁)) ∈ (ℝ ∖ ℚ)) |
8 | 7 | eldifbd 3949 | . . 3 ⊢ (𝜑 → ¬ (2↑𝑐(1 / 𝑁)) ∈ ℚ) |
9 | fltne.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
10 | 9 | nnzd 12087 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℤ) |
11 | fltne.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
12 | znq 12353 | . . . . . 6 ⊢ ((𝐶 ∈ ℤ ∧ 𝐴 ∈ ℕ) → (𝐶 / 𝐴) ∈ ℚ) | |
13 | 10, 11, 12 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝐶 / 𝐴) ∈ ℚ) |
14 | eleq1a 2908 | . . . . 5 ⊢ ((𝐶 / 𝐴) ∈ ℚ → ((2↑𝑐(1 / 𝑁)) = (𝐶 / 𝐴) → (2↑𝑐(1 / 𝑁)) ∈ ℚ)) | |
15 | 13, 14 | syl 17 | . . . 4 ⊢ (𝜑 → ((2↑𝑐(1 / 𝑁)) = (𝐶 / 𝐴) → (2↑𝑐(1 / 𝑁)) ∈ ℚ)) |
16 | 15 | necon3bd 3030 | . . 3 ⊢ (𝜑 → (¬ (2↑𝑐(1 / 𝑁)) ∈ ℚ → (2↑𝑐(1 / 𝑁)) ≠ (𝐶 / 𝐴))) |
17 | 8, 16 | mpd 15 | . 2 ⊢ (𝜑 → (2↑𝑐(1 / 𝑁)) ≠ (𝐶 / 𝐴)) |
18 | 2rp 12395 | . . . . . 6 ⊢ 2 ∈ ℝ+ | |
19 | 18 | a1i 11 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℝ+) |
20 | eluzge3nn 12291 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℕ) | |
21 | 3, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
22 | 21 | nnrecred 11689 | . . . . 5 ⊢ (𝜑 → (1 / 𝑁) ∈ ℝ) |
23 | 19, 22 | rpcxpcld 25315 | . . . 4 ⊢ (𝜑 → (2↑𝑐(1 / 𝑁)) ∈ ℝ+) |
24 | 23 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (2↑𝑐(1 / 𝑁)) ∈ ℝ+) |
25 | 9 | nnrpd 12430 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
26 | 11 | nnrpd 12430 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
27 | 25, 26 | rpdivcld 12449 | . . . 4 ⊢ (𝜑 → (𝐶 / 𝐴) ∈ ℝ+) |
28 | 27 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐶 / 𝐴) ∈ ℝ+) |
29 | simpl 485 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝜑) | |
30 | eluzelz 12254 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘3) → 𝑁 ∈ ℤ) | |
31 | 29, 3, 30 | 3syl 18 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝑁 ∈ ℤ) |
32 | 21 | nnne0d 11688 | . . . 4 ⊢ (𝜑 → 𝑁 ≠ 0) |
33 | 32 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝑁 ≠ 0) |
34 | 11 | nncnd 11654 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
35 | 21 | nnnn0d 11956 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
36 | 34, 35 | expcld 13511 | . . . . . 6 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
37 | 36 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴↑𝑁) ∈ ℂ) |
38 | 2cnd 11716 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 2 ∈ ℂ) | |
39 | 11 | nnne0d 11688 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≠ 0) |
40 | 3, 30 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
41 | 34, 39, 40 | expne0d 13517 | . . . . . 6 ⊢ (𝜑 → (𝐴↑𝑁) ≠ 0) |
42 | 41 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴↑𝑁) ≠ 0) |
43 | 37 | times2d 11882 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ((𝐴↑𝑁) · 2) = ((𝐴↑𝑁) + (𝐴↑𝑁))) |
44 | simpr 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵) | |
45 | 44 | oveq1d 7171 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴↑𝑁) = (𝐵↑𝑁)) |
46 | 45 | oveq2d 7172 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ((𝐴↑𝑁) + (𝐴↑𝑁)) = ((𝐴↑𝑁) + (𝐵↑𝑁))) |
47 | fltne.1 | . . . . . . 7 ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) | |
48 | 47 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) |
49 | 43, 46, 48 | 3eqtrd 2860 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ((𝐴↑𝑁) · 2) = (𝐶↑𝑁)) |
50 | 37, 38, 42, 49 | mvllmuld 11472 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 2 = ((𝐶↑𝑁) / (𝐴↑𝑁))) |
51 | 21 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝑁 ∈ ℕ) |
52 | cxproot 25273 | . . . . 5 ⊢ ((2 ∈ ℂ ∧ 𝑁 ∈ ℕ) → ((2↑𝑐(1 / 𝑁))↑𝑁) = 2) | |
53 | 38, 51, 52 | syl2anc 586 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ((2↑𝑐(1 / 𝑁))↑𝑁) = 2) |
54 | 9 | nncnd 11654 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
55 | 54, 34, 39, 35 | expdivd 13525 | . . . . 5 ⊢ (𝜑 → ((𝐶 / 𝐴)↑𝑁) = ((𝐶↑𝑁) / (𝐴↑𝑁))) |
56 | 55 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ((𝐶 / 𝐴)↑𝑁) = ((𝐶↑𝑁) / (𝐴↑𝑁))) |
57 | 50, 53, 56 | 3eqtr4d 2866 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ((2↑𝑐(1 / 𝑁))↑𝑁) = ((𝐶 / 𝐴)↑𝑁)) |
58 | 24, 28, 31, 33, 57 | exp11d 39238 | . 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (2↑𝑐(1 / 𝑁)) = (𝐶 / 𝐴)) |
59 | 17, 58 | mteqand 3122 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3933 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 / cdiv 11297 ℕcn 11638 2c2 11693 3c3 11694 ℤcz 11982 ℤ≥cuz 12244 ℚcq 12349 ℝ+crp 12390 ↑cexp 13430 ℙcprime 16015 ↑𝑐ccxp 25139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ioc 12744 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 df-sin 15423 df-cos 15424 df-pi 15426 df-dvds 15608 df-gcd 15844 df-prm 16016 df-numer 16075 df-denom 16076 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 df-log 25140 df-cxp 25141 |
This theorem is referenced by: (None) |
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