Theorem pceulem 12463

Description: Lemma for pceu 12464. (Contributed by Mario Carneiro, 23-Feb-2014.)

Hypotheses

Ref	Expression
pcval.1	|- S = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )
pcval.2	|- T = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )
pceu.3	|- U = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < )
pceu.4	|- V = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < )
pceu.5	|- ( ph -> P e. Prime )
pceu.6	|- ( ph -> N =/= 0 )
pceu.7	|- ( ph -> ( x e. ZZ /\ y e. NN ) )
pceu.8	|- ( ph -> N = ( x / y ) )
pceu.9	|- ( ph -> ( s e. ZZ /\ t e. NN ) )
pceu.10	|- ( ph -> N = ( s / t ) )

Assertion

Ref	Expression
pceulem	|- ( ph -> ( S - T ) = ( U - V ) )

Distinct variable groups:   n, s, t, x, y, N   P, n, s, t, x, y   S, s, t   T, s, t

Allowed substitution hints:   ph( x, y, t, n, s)   S( x, y, n)   T( x, y, n)   U( x, y, t, n, s)   V( x, y, t, n, s)

Proof of Theorem pceulem

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pceu.7	. . . . . . . . . . 11 |- ( ph -> ( x e. ZZ /\ y e. NN ) )
2	1	simprd 114	. . . . . . . . . 10 |- ( ph -> y e. NN )
3	2	nncnd 9004	. . . . . . . . 9 |- ( ph -> y e. CC )
4		pceu.9	. . . . . . . . . . 11 |- ( ph -> ( s e. ZZ /\ t e. NN ) )
5	4	simpld 112	. . . . . . . . . 10 |- ( ph -> s e. ZZ )
6	5	zcnd 9449	. . . . . . . . 9 |- ( ph -> s e. CC )
7	3, 6	mulcomd 8048	. . . . . . . 8 |- ( ph -> ( y x. s ) = ( s x. y ) )
8		pceu.10	. . . . . . . . . 10 |- ( ph -> N = ( s / t ) )
9		pceu.8	. . . . . . . . . 10 |- ( ph -> N = ( x / y ) )
10	8, 9	eqtr3d 2231	. . . . . . . . 9 |- ( ph -> ( s / t ) = ( x / y ) )
11	4	simprd 114	. . . . . . . . . . 11 |- ( ph -> t e. NN )
12	11	nncnd 9004	. . . . . . . . . 10 |- ( ph -> t e. CC )
13	1	simpld 112	. . . . . . . . . . 11 |- ( ph -> x e. ZZ )
14	13	zcnd 9449	. . . . . . . . . 10 |- ( ph -> x e. CC )
15	11	nnap0d 9036	. . . . . . . . . 10 |- ( ph -> t # 0 )
16	2	nnap0d 9036	. . . . . . . . . 10 |- ( ph -> y # 0 )
17	6, 12, 14, 3, 15, 16	divmuleqapd 8860	. . . . . . . . 9 |- ( ph -> ( ( s / t ) = ( x / y ) <-> ( s x. y ) = ( x x. t ) ) )
18	10, 17	mpbid 147	. . . . . . . 8 |- ( ph -> ( s x. y ) = ( x x. t ) )
19	7, 18	eqtrd 2229	. . . . . . 7 |- ( ph -> ( y x. s ) = ( x x. t ) )
20	19	breq2d 4045	. . . . . 6 |- ( ph -> ( ( P ^ z ) || ( y x. s ) <-> ( P ^ z ) || ( x x. t ) ) )
21	20	rabbidv 2752	. . . . 5 |- ( ph -> { z e. NN0 | ( P ^ z ) || ( y x. s ) } = { z e. NN0 | ( P ^ z ) || ( x x. t ) } )
22		oveq2 5930	. . . . . . 7 |- ( n = z -> ( P ^ n ) = ( P ^ z ) )
23	22	breq1d 4043	. . . . . 6 |- ( n = z -> ( ( P ^ n ) || ( y x. s ) <-> ( P ^ z ) || ( y x. s ) ) )
24	23	cbvrabv 2762	. . . . 5 |- { n e. NN0 | ( P ^ n ) || ( y x. s ) } = { z e. NN0 | ( P ^ z ) || ( y x. s ) }
25	22	breq1d 4043	. . . . . 6 |- ( n = z -> ( ( P ^ n ) || ( x x. t ) <-> ( P ^ z ) || ( x x. t ) ) )
26	25	cbvrabv 2762	. . . . 5 |- { n e. NN0 | ( P ^ n ) || ( x x. t ) } = { z e. NN0 | ( P ^ z ) || ( x x. t ) }
27	21, 24, 26	3eqtr4g 2254	. . . 4 |- ( ph -> { n e. NN0 | ( P ^ n ) || ( y x. s ) } = { n e. NN0 | ( P ^ n ) || ( x x. t ) } )
28	27	supeq1d 7053	. . 3 |- ( ph -> sup ( { n e. NN0 | ( P ^ n ) || ( y x. s ) } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || ( x x. t ) } , RR , < ) )
29		pceu.5	. . . 4 |- ( ph -> P e. Prime )
30	2	nnzd 9447	. . . 4 |- ( ph -> y e. ZZ )
31	2	nnne0d 9035	. . . 4 |- ( ph -> y =/= 0 )
32		pceu.6	. . . . 5 |- ( ph -> N =/= 0 )
33	12, 15	div0apd 8814	. . . . . . . 8 |- ( ph -> ( 0 / t ) = 0 )
34		oveq1 5929	. . . . . . . . 9 |- ( s = 0 -> ( s / t ) = ( 0 / t ) )
35	34	eqeq1d 2205	. . . . . . . 8 |- ( s = 0 -> ( ( s / t ) = 0 <-> ( 0 / t ) = 0 ) )
36	33, 35	syl5ibrcom 157	. . . . . . 7 |- ( ph -> ( s = 0 -> ( s / t ) = 0 ) )
37	8	eqeq1d 2205	. . . . . . 7 |- ( ph -> ( N = 0 <-> ( s / t ) = 0 ) )
38	36, 37	sylibrd 169	. . . . . 6 |- ( ph -> ( s = 0 -> N = 0 ) )
39	38	necon3d 2411	. . . . 5 |- ( ph -> ( N =/= 0 -> s =/= 0 ) )
40	32, 39	mpd 13	. . . 4 |- ( ph -> s =/= 0 )
41		pcval.2	. . . . 5 |- T = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )
42		pceu.3	. . . . 5 |- U = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < )
43		eqid 2196	. . . . 5 |- sup ( { n e. NN0 | ( P ^ n ) || ( y x. s ) } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || ( y x. s ) } , RR , < )
44	41, 42, 43	pcpremul 12462	. . . 4 |- ( ( P e. Prime /\ ( y e. ZZ /\ y =/= 0 ) /\ ( s e. ZZ /\ s =/= 0 ) ) -> ( T + U ) = sup ( { n e. NN0 | ( P ^ n ) || ( y x. s ) } , RR , < ) )
45	29, 30, 31, 5, 40, 44	syl122anc 1258	. . 3 |- ( ph -> ( T + U ) = sup ( { n e. NN0 | ( P ^ n ) || ( y x. s ) } , RR , < ) )
46	3, 16	div0apd 8814	. . . . . . . 8 |- ( ph -> ( 0 / y ) = 0 )
47		oveq1 5929	. . . . . . . . 9 |- ( x = 0 -> ( x / y ) = ( 0 / y ) )
48	47	eqeq1d 2205	. . . . . . . 8 |- ( x = 0 -> ( ( x / y ) = 0 <-> ( 0 / y ) = 0 ) )
49	46, 48	syl5ibrcom 157	. . . . . . 7 |- ( ph -> ( x = 0 -> ( x / y ) = 0 ) )
50	9	eqeq1d 2205	. . . . . . 7 |- ( ph -> ( N = 0 <-> ( x / y ) = 0 ) )
51	49, 50	sylibrd 169	. . . . . 6 |- ( ph -> ( x = 0 -> N = 0 ) )
52	51	necon3d 2411	. . . . 5 |- ( ph -> ( N =/= 0 -> x =/= 0 ) )
53	32, 52	mpd 13	. . . 4 |- ( ph -> x =/= 0 )
54	11	nnzd 9447	. . . 4 |- ( ph -> t e. ZZ )
55	11	nnne0d 9035	. . . 4 |- ( ph -> t =/= 0 )
56		pcval.1	. . . . 5 |- S = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )
57		pceu.4	. . . . 5 |- V = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < )
58		eqid 2196	. . . . 5 |- sup ( { n e. NN0 | ( P ^ n ) || ( x x. t ) } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || ( x x. t ) } , RR , < )
59	56, 57, 58	pcpremul 12462	. . . 4 |- ( ( P e. Prime /\ ( x e. ZZ /\ x =/= 0 ) /\ ( t e. ZZ /\ t =/= 0 ) ) -> ( S + V ) = sup ( { n e. NN0 | ( P ^ n ) || ( x x. t ) } , RR , < ) )
60	29, 13, 53, 54, 55, 59	syl122anc 1258	. . 3 |- ( ph -> ( S + V ) = sup ( { n e. NN0 | ( P ^ n ) || ( x x. t ) } , RR , < ) )
61	28, 45, 60	3eqtr4d 2239	. 2 |- ( ph -> ( T + U ) = ( S + V ) )
62		prmuz2 12299	. . . . . 6 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
63	29, 62	syl 14	. . . . 5 |- ( ph -> P e. ( ZZ>= ` 2 ) )
64		eqid 2196	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || y }
65	64, 41	pcprecl 12458	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( y e. ZZ /\ y =/= 0 ) ) -> ( T e. NN0 /\ ( P ^ T ) || y ) )
66	65	simpld 112	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( y e. ZZ /\ y =/= 0 ) ) -> T e. NN0 )
67	63, 30, 31, 66	syl12anc 1247	. . . 4 |- ( ph -> T e. NN0 )
68	67	nn0cnd 9304	. . 3 |- ( ph -> T e. CC )
69		eqid 2196	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || s } = { n e. NN0 | ( P ^ n ) || s }
70	69, 42	pcprecl 12458	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( s e. ZZ /\ s =/= 0 ) ) -> ( U e. NN0 /\ ( P ^ U ) || s ) )
71	70	simpld 112	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( s e. ZZ /\ s =/= 0 ) ) -> U e. NN0 )
72	63, 5, 40, 71	syl12anc 1247	. . . 4 |- ( ph -> U e. NN0 )
73	72	nn0cnd 9304	. . 3 |- ( ph -> U e. CC )
74		eqid 2196	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || x }
75	74, 56	pcprecl 12458	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || x ) )
76	75	simpld 112	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( x e. ZZ /\ x =/= 0 ) ) -> S e. NN0 )
77	63, 13, 53, 76	syl12anc 1247	. . . 4 |- ( ph -> S e. NN0 )
78	77	nn0cnd 9304	. . 3 |- ( ph -> S e. CC )
79		eqid 2196	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || t } = { n e. NN0 | ( P ^ n ) || t }
80	79, 57	pcprecl 12458	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( t e. ZZ /\ t =/= 0 ) ) -> ( V e. NN0 /\ ( P ^ V ) || t ) )
81	80	simpld 112	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( t e. ZZ /\ t =/= 0 ) ) -> V e. NN0 )
82	63, 54, 55, 81	syl12anc 1247	. . . 4 |- ( ph -> V e. NN0 )
83	82	nn0cnd 9304	. . 3 |- ( ph -> V e. CC )
84	68, 73, 78, 83	addsubeq4d 8388	. 2 |- ( ph -> ( ( T + U ) = ( S + V ) <-> ( S - T ) = ( U - V ) ) )
85	61, 84	mpbid 147	1 |- ( ph -> ( S - T ) = ( U - V ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   =/= wne 2367   {crab 2479   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   supcsup 7048   RRcr 7878   0cc0 7879   + caddc 7882   x. cmul 7884   < clt 8061   - cmin 8197   / cdiv 8699   NNcn 8990   2c2 9041   NN0cn0 9249   ZZcz 9326   ZZ>=cuz 9601   ^cexp 10630   || cdvds 11952   Primecprime 12275

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-1o 6474  df-2o 6475  df-er 6592  df-en 6800  df-sup 7050  df-inf 7051  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fz 10084  df-fzo 10218  df-fl 10360  df-mod 10415  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164  df-dvds 11953  df-gcd 12121  df-prm 12276

This theorem is referenced by:  pceu  12464