Theorem pceulem 12296

Description: Lemma for pceu 12297. (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 8935	. . . . . . . . 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 9378	. . . . . . . . 9 |- ( ph -> s e. CC )
7	3, 6	mulcomd 7981	. . . . . . . 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 2212	. . . . . . . . 9 |- ( ph -> ( s / t ) = ( x / y ) )
11	4	simprd 114	. . . . . . . . . . 11 |- ( ph -> t e. NN )
12	11	nncnd 8935	. . . . . . . . . 10 |- ( ph -> t e. CC )
13	1	simpld 112	. . . . . . . . . . 11 |- ( ph -> x e. ZZ )
14	13	zcnd 9378	. . . . . . . . . 10 |- ( ph -> x e. CC )
15	11	nnap0d 8967	. . . . . . . . . 10 |- ( ph -> t # 0 )
16	2	nnap0d 8967	. . . . . . . . . 10 |- ( ph -> y # 0 )
17	6, 12, 14, 3, 15, 16	divmuleqapd 8792	. . . . . . . . 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 2210	. . . . . . 7 |- ( ph -> ( y x. s ) = ( x x. t ) )
20	19	breq2d 4017	. . . . . 6 |- ( ph -> ( ( P ^ z ) || ( y x. s ) <-> ( P ^ z ) || ( x x. t ) ) )
21	20	rabbidv 2728	. . . . 5 |- ( ph -> { z e. NN0 | ( P ^ z ) || ( y x. s ) } = { z e. NN0 | ( P ^ z ) || ( x x. t ) } )
22		oveq2 5885	. . . . . . 7 |- ( n = z -> ( P ^ n ) = ( P ^ z ) )
23	22	breq1d 4015	. . . . . 6 |- ( n = z -> ( ( P ^ n ) || ( y x. s ) <-> ( P ^ z ) || ( y x. s ) ) )
24	23	cbvrabv 2738	. . . . 5 |- { n e. NN0 | ( P ^ n ) || ( y x. s ) } = { z e. NN0 | ( P ^ z ) || ( y x. s ) }
25	22	breq1d 4015	. . . . . 6 |- ( n = z -> ( ( P ^ n ) || ( x x. t ) <-> ( P ^ z ) || ( x x. t ) ) )
26	25	cbvrabv 2738	. . . . 5 |- { n e. NN0 | ( P ^ n ) || ( x x. t ) } = { z e. NN0 | ( P ^ z ) || ( x x. t ) }
27	21, 24, 26	3eqtr4g 2235	. . . 4 |- ( ph -> { n e. NN0 | ( P ^ n ) || ( y x. s ) } = { n e. NN0 | ( P ^ n ) || ( x x. t ) } )
28	27	supeq1d 6988	. . 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 9376	. . . 4 |- ( ph -> y e. ZZ )
31	2	nnne0d 8966	. . . 4 |- ( ph -> y =/= 0 )
32		pceu.6	. . . . 5 |- ( ph -> N =/= 0 )
33	12, 15	div0apd 8746	. . . . . . . 8 |- ( ph -> ( 0 / t ) = 0 )
34		oveq1 5884	. . . . . . . . 9 |- ( s = 0 -> ( s / t ) = ( 0 / t ) )
35	34	eqeq1d 2186	. . . . . . . 8 |- ( s = 0 -> ( ( s / t ) = 0 <-> ( 0 / t ) = 0 ) )
36	33, 35	syl5ibrcom 157	. . . . . . 7 |- ( ph -> ( s = 0 -> ( s / t ) = 0 ) )
37	8	eqeq1d 2186	. . . . . . 7 |- ( ph -> ( N = 0 <-> ( s / t ) = 0 ) )
38	36, 37	sylibrd 169	. . . . . 6 |- ( ph -> ( s = 0 -> N = 0 ) )
39	38	necon3d 2391	. . . . 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 2177	. . . . 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 12295	. . . 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 1247	. . 3 |- ( ph -> ( T + U ) = sup ( { n e. NN0 | ( P ^ n ) || ( y x. s ) } , RR , < ) )
46	3, 16	div0apd 8746	. . . . . . . 8 |- ( ph -> ( 0 / y ) = 0 )
47		oveq1 5884	. . . . . . . . 9 |- ( x = 0 -> ( x / y ) = ( 0 / y ) )
48	47	eqeq1d 2186	. . . . . . . 8 |- ( x = 0 -> ( ( x / y ) = 0 <-> ( 0 / y ) = 0 ) )
49	46, 48	syl5ibrcom 157	. . . . . . 7 |- ( ph -> ( x = 0 -> ( x / y ) = 0 ) )
50	9	eqeq1d 2186	. . . . . . 7 |- ( ph -> ( N = 0 <-> ( x / y ) = 0 ) )
51	49, 50	sylibrd 169	. . . . . 6 |- ( ph -> ( x = 0 -> N = 0 ) )
52	51	necon3d 2391	. . . . 5 |- ( ph -> ( N =/= 0 -> x =/= 0 ) )
53	32, 52	mpd 13	. . . 4 |- ( ph -> x =/= 0 )
54	11	nnzd 9376	. . . 4 |- ( ph -> t e. ZZ )
55	11	nnne0d 8966	. . . 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 2177	. . . . 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 12295	. . . 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 1247	. . 3 |- ( ph -> ( S + V ) = sup ( { n e. NN0 | ( P ^ n ) || ( x x. t ) } , RR , < ) )
61	28, 45, 60	3eqtr4d 2220	. 2 |- ( ph -> ( T + U ) = ( S + V ) )
62		prmuz2 12133	. . . . . 6 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
63	29, 62	syl 14	. . . . 5 |- ( ph -> P e. ( ZZ>= ` 2 ) )
64		eqid 2177	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || y }
65	64, 41	pcprecl 12291	. . . . . 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 1236	. . . 4 |- ( ph -> T e. NN0 )
68	67	nn0cnd 9233	. . 3 |- ( ph -> T e. CC )
69		eqid 2177	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || s } = { n e. NN0 | ( P ^ n ) || s }
70	69, 42	pcprecl 12291	. . . . . 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 1236	. . . 4 |- ( ph -> U e. NN0 )
73	72	nn0cnd 9233	. . 3 |- ( ph -> U e. CC )
74		eqid 2177	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || x }
75	74, 56	pcprecl 12291	. . . . . 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 1236	. . . 4 |- ( ph -> S e. NN0 )
78	77	nn0cnd 9233	. . 3 |- ( ph -> S e. CC )
79		eqid 2177	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || t } = { n e. NN0 | ( P ^ n ) || t }
80	79, 57	pcprecl 12291	. . . . . 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 1236	. . . 4 |- ( ph -> V e. NN0 )
83	82	nn0cnd 9233	. . 3 |- ( ph -> V e. CC )
84	68, 73, 78, 83	addsubeq4d 8321	. 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 1353   e. wcel 2148   =/= wne 2347   {crab 2459   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   supcsup 6983   RRcr 7812   0cc0 7813   + caddc 7816   x. cmul 7818   < clt 7994   - cmin 8130   / cdiv 8631   NNcn 8921   2c2 8972   NN0cn0 9178   ZZcz 9255   ZZ>=cuz 9530   ^cexp 10521   || cdvds 11796   Primecprime 12109

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-1o 6419  df-2o 6420  df-er 6537  df-en 6743  df-sup 6985  df-inf 6986  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fz 10011  df-fzo 10145  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-dvds 11797  df-gcd 11946  df-prm 12110

This theorem is referenced by:  pceu  12297