Metamath Proof Explorer


Theorem fltnlta

Description: In a Fermat counterexample, the exponent N is less than all three numbers ( A , B , and C ). Note that A < B (hypothesis) and B < C ( fltltc ). See https://youtu.be/EymVXkPWxyc for an outline. (Contributed by SN, 24-Aug-2023)

Ref Expression
Hypotheses fltltc.a
|- ( ph -> A e. NN )
fltltc.b
|- ( ph -> B e. NN )
fltltc.c
|- ( ph -> C e. NN )
fltltc.n
|- ( ph -> N e. ( ZZ>= ` 3 ) )
fltltc.1
|- ( ph -> ( ( A ^ N ) + ( B ^ N ) ) = ( C ^ N ) )
fltnlta.1
|- ( ph -> A < B )
Assertion fltnlta
|- ( ph -> N < A )

Proof

Step Hyp Ref Expression
1 fltltc.a
 |-  ( ph -> A e. NN )
2 fltltc.b
 |-  ( ph -> B e. NN )
3 fltltc.c
 |-  ( ph -> C e. NN )
4 fltltc.n
 |-  ( ph -> N e. ( ZZ>= ` 3 ) )
5 fltltc.1
 |-  ( ph -> ( ( A ^ N ) + ( B ^ N ) ) = ( C ^ N ) )
6 fltnlta.1
 |-  ( ph -> A < B )
7 eluzge3nn
 |-  ( N e. ( ZZ>= ` 3 ) -> N e. NN )
8 4 7 syl
 |-  ( ph -> N e. NN )
9 8 nnred
 |-  ( ph -> N e. RR )
10 3 nnred
 |-  ( ph -> C e. RR )
11 2 nnred
 |-  ( ph -> B e. RR )
12 10 11 resubcld
 |-  ( ph -> ( C - B ) e. RR )
13 uzuzle23
 |-  ( N e. ( ZZ>= ` 3 ) -> N e. ( ZZ>= ` 2 ) )
14 uz2m1nn
 |-  ( N e. ( ZZ>= ` 2 ) -> ( N - 1 ) e. NN )
15 4 13 14 3syl
 |-  ( ph -> ( N - 1 ) e. NN )
16 15 nnnn0d
 |-  ( ph -> ( N - 1 ) e. NN0 )
17 10 16 reexpcld
 |-  ( ph -> ( C ^ ( N - 1 ) ) e. RR )
18 15 nnred
 |-  ( ph -> ( N - 1 ) e. RR )
19 11 16 reexpcld
 |-  ( ph -> ( B ^ ( N - 1 ) ) e. RR )
20 18 19 remulcld
 |-  ( ph -> ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) e. RR )
21 17 20 readdcld
 |-  ( ph -> ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) e. RR )
22 12 21 remulcld
 |-  ( ph -> ( ( C - B ) x. ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) e. RR )
23 1 nnrpd
 |-  ( ph -> A e. RR+ )
24 15 nnzd
 |-  ( ph -> ( N - 1 ) e. ZZ )
25 23 24 rpexpcld
 |-  ( ph -> ( A ^ ( N - 1 ) ) e. RR+ )
26 22 25 rerpdivcld
 |-  ( ph -> ( ( ( C - B ) x. ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) / ( A ^ ( N - 1 ) ) ) e. RR )
27 1 nnred
 |-  ( ph -> A e. RR )
28 19 20 readdcld
 |-  ( ph -> ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) e. RR )
29 12 28 remulcld
 |-  ( ph -> ( ( C - B ) x. ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) e. RR )
30 29 25 rerpdivcld
 |-  ( ph -> ( ( ( C - B ) x. ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) / ( A ^ ( N - 1 ) ) ) e. RR )
31 12 9 remulcld
 |-  ( ph -> ( ( C - B ) x. N ) e. RR )
32 1cnd
 |-  ( ph -> 1 e. CC )
33 15 nncnd
 |-  ( ph -> ( N - 1 ) e. CC )
34 19 recnd
 |-  ( ph -> ( B ^ ( N - 1 ) ) e. CC )
35 32 33 34 adddird
 |-  ( ph -> ( ( 1 + ( N - 1 ) ) x. ( B ^ ( N - 1 ) ) ) = ( ( 1 x. ( B ^ ( N - 1 ) ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) )
36 8 nncnd
 |-  ( ph -> N e. CC )
37 32 36 pncan3d
 |-  ( ph -> ( 1 + ( N - 1 ) ) = N )
38 37 oveq1d
 |-  ( ph -> ( ( 1 + ( N - 1 ) ) x. ( B ^ ( N - 1 ) ) ) = ( N x. ( B ^ ( N - 1 ) ) ) )
39 34 mulid2d
 |-  ( ph -> ( 1 x. ( B ^ ( N - 1 ) ) ) = ( B ^ ( N - 1 ) ) )
40 39 oveq1d
 |-  ( ph -> ( ( 1 x. ( B ^ ( N - 1 ) ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) = ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) )
41 35 38 40 3eqtr3rd
 |-  ( ph -> ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) = ( N x. ( B ^ ( N - 1 ) ) ) )
42 41 oveq2d
 |-  ( ph -> ( ( C - B ) x. ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) = ( ( C - B ) x. ( N x. ( B ^ ( N - 1 ) ) ) ) )
43 42 oveq1d
 |-  ( ph -> ( ( ( C - B ) x. ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) / ( A ^ ( N - 1 ) ) ) = ( ( ( C - B ) x. ( N x. ( B ^ ( N - 1 ) ) ) ) / ( A ^ ( N - 1 ) ) ) )
44 43 30 eqeltrrd
 |-  ( ph -> ( ( ( C - B ) x. ( N x. ( B ^ ( N - 1 ) ) ) ) / ( A ^ ( N - 1 ) ) ) e. RR )
45 8 nnnn0d
 |-  ( ph -> N e. NN0 )
46 45 nn0ge0d
 |-  ( ph -> 0 <_ N )
47 1red
 |-  ( ph -> 1 e. RR )
48 1 2 3 4 5 fltltc
 |-  ( ph -> B < C )
49 nnltp1le
 |-  ( ( B e. NN /\ C e. NN ) -> ( B < C <-> ( B + 1 ) <_ C ) )
50 2 3 49 syl2anc
 |-  ( ph -> ( B < C <-> ( B + 1 ) <_ C ) )
51 48 50 mpbid
 |-  ( ph -> ( B + 1 ) <_ C )
52 11 leidd
 |-  ( ph -> B <_ B )
53 10 11 47 11 51 52 lesub3d
 |-  ( ph -> 1 <_ ( C - B ) )
54 9 12 46 53 lemulge12d
 |-  ( ph -> N <_ ( ( C - B ) x. N ) )
55 12 recnd
 |-  ( ph -> ( C - B ) e. CC )
56 25 rpred
 |-  ( ph -> ( A ^ ( N - 1 ) ) e. RR )
57 56 recnd
 |-  ( ph -> ( A ^ ( N - 1 ) ) e. CC )
58 55 36 57 mulassd
 |-  ( ph -> ( ( ( C - B ) x. N ) x. ( A ^ ( N - 1 ) ) ) = ( ( C - B ) x. ( N x. ( A ^ ( N - 1 ) ) ) ) )
59 58 oveq1d
 |-  ( ph -> ( ( ( ( C - B ) x. N ) x. ( A ^ ( N - 1 ) ) ) / ( A ^ ( N - 1 ) ) ) = ( ( ( C - B ) x. ( N x. ( A ^ ( N - 1 ) ) ) ) / ( A ^ ( N - 1 ) ) ) )
60 55 36 mulcld
 |-  ( ph -> ( ( C - B ) x. N ) e. CC )
61 1 nncnd
 |-  ( ph -> A e. CC )
62 1 nnne0d
 |-  ( ph -> A =/= 0 )
63 61 62 24 expne0d
 |-  ( ph -> ( A ^ ( N - 1 ) ) =/= 0 )
64 60 57 63 divcan4d
 |-  ( ph -> ( ( ( ( C - B ) x. N ) x. ( A ^ ( N - 1 ) ) ) / ( A ^ ( N - 1 ) ) ) = ( ( C - B ) x. N ) )
65 59 64 eqtr3d
 |-  ( ph -> ( ( ( C - B ) x. ( N x. ( A ^ ( N - 1 ) ) ) ) / ( A ^ ( N - 1 ) ) ) = ( ( C - B ) x. N ) )
66 9 56 remulcld
 |-  ( ph -> ( N x. ( A ^ ( N - 1 ) ) ) e. RR )
67 12 66 remulcld
 |-  ( ph -> ( ( C - B ) x. ( N x. ( A ^ ( N - 1 ) ) ) ) e. RR )
68 42 29 eqeltrrd
 |-  ( ph -> ( ( C - B ) x. ( N x. ( B ^ ( N - 1 ) ) ) ) e. RR )
69 41 28 eqeltrrd
 |-  ( ph -> ( N x. ( B ^ ( N - 1 ) ) ) e. RR )
70 difrp
 |-  ( ( B e. RR /\ C e. RR ) -> ( B < C <-> ( C - B ) e. RR+ ) )
71 11 10 70 syl2anc
 |-  ( ph -> ( B < C <-> ( C - B ) e. RR+ ) )
72 48 71 mpbid
 |-  ( ph -> ( C - B ) e. RR+ )
73 8 nnrpd
 |-  ( ph -> N e. RR+ )
74 2 nnrpd
 |-  ( ph -> B e. RR+ )
75 23 74 15 6 ltexp1dd
 |-  ( ph -> ( A ^ ( N - 1 ) ) < ( B ^ ( N - 1 ) ) )
76 56 19 73 75 ltmul2dd
 |-  ( ph -> ( N x. ( A ^ ( N - 1 ) ) ) < ( N x. ( B ^ ( N - 1 ) ) ) )
77 66 69 72 76 ltmul2dd
 |-  ( ph -> ( ( C - B ) x. ( N x. ( A ^ ( N - 1 ) ) ) ) < ( ( C - B ) x. ( N x. ( B ^ ( N - 1 ) ) ) ) )
78 67 68 25 77 ltdiv1dd
 |-  ( ph -> ( ( ( C - B ) x. ( N x. ( A ^ ( N - 1 ) ) ) ) / ( A ^ ( N - 1 ) ) ) < ( ( ( C - B ) x. ( N x. ( B ^ ( N - 1 ) ) ) ) / ( A ^ ( N - 1 ) ) ) )
79 65 78 eqbrtrrd
 |-  ( ph -> ( ( C - B ) x. N ) < ( ( ( C - B ) x. ( N x. ( B ^ ( N - 1 ) ) ) ) / ( A ^ ( N - 1 ) ) ) )
80 9 31 44 54 79 lelttrd
 |-  ( ph -> N < ( ( ( C - B ) x. ( N x. ( B ^ ( N - 1 ) ) ) ) / ( A ^ ( N - 1 ) ) ) )
81 80 43 breqtrrd
 |-  ( ph -> N < ( ( ( C - B ) x. ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) / ( A ^ ( N - 1 ) ) ) )
82 3 nnrpd
 |-  ( ph -> C e. RR+ )
83 74 82 15 48 ltexp1dd
 |-  ( ph -> ( B ^ ( N - 1 ) ) < ( C ^ ( N - 1 ) ) )
84 19 17 20 83 ltadd1dd
 |-  ( ph -> ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) < ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) )
85 28 21 72 84 ltmul2dd
 |-  ( ph -> ( ( C - B ) x. ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) < ( ( C - B ) x. ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) )
86 29 22 25 85 ltdiv1dd
 |-  ( ph -> ( ( ( C - B ) x. ( ( B ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) / ( A ^ ( N - 1 ) ) ) < ( ( ( C - B ) x. ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) / ( A ^ ( N - 1 ) ) ) )
87 9 30 26 81 86 lttrd
 |-  ( ph -> N < ( ( ( C - B ) x. ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) / ( A ^ ( N - 1 ) ) ) )
88 27 45 reexpcld
 |-  ( ph -> ( A ^ N ) e. RR )
89 1 2 3 4 5 fltnltalem
 |-  ( ph -> ( ( C - B ) x. ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) < ( A ^ N ) )
90 22 88 25 89 ltdiv1dd
 |-  ( ph -> ( ( ( C - B ) x. ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) / ( A ^ ( N - 1 ) ) ) < ( ( A ^ N ) / ( A ^ ( N - 1 ) ) ) )
91 36 32 nncand
 |-  ( ph -> ( N - ( N - 1 ) ) = 1 )
92 91 oveq2d
 |-  ( ph -> ( A ^ ( N - ( N - 1 ) ) ) = ( A ^ 1 ) )
93 8 nnzd
 |-  ( ph -> N e. ZZ )
94 61 62 24 93 expsubd
 |-  ( ph -> ( A ^ ( N - ( N - 1 ) ) ) = ( ( A ^ N ) / ( A ^ ( N - 1 ) ) ) )
95 61 exp1d
 |-  ( ph -> ( A ^ 1 ) = A )
96 92 94 95 3eqtr3d
 |-  ( ph -> ( ( A ^ N ) / ( A ^ ( N - 1 ) ) ) = A )
97 90 96 breqtrd
 |-  ( ph -> ( ( ( C - B ) x. ( ( C ^ ( N - 1 ) ) + ( ( N - 1 ) x. ( B ^ ( N - 1 ) ) ) ) ) / ( A ^ ( N - 1 ) ) ) < A )
98 9 26 27 87 97 lttrd
 |-  ( ph -> N < A )