Theorem pceulem 15550

Description: Lemma for pceu 15551. (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 479	. . . . . . . . . 10 |- ( ph -> y e. NN )
3	2	nncnd 11036	. . . . . . . . 9 |- ( ph -> y e. CC )
4		pceu.9	. . . . . . . . . . 11 |- ( ph -> ( s e. ZZ /\ t e. NN ) )
5	4	simpld 475	. . . . . . . . . 10 |- ( ph -> s e. ZZ )
6	5	zcnd 11483	. . . . . . . . 9 |- ( ph -> s e. CC )
7	3, 6	mulcomd 10061	. . . . . . . 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 2658	. . . . . . . . 9 |- ( ph -> ( s / t ) = ( x / y ) )
11	4	simprd 479	. . . . . . . . . . 11 |- ( ph -> t e. NN )
12	11	nncnd 11036	. . . . . . . . . 10 |- ( ph -> t e. CC )
13	1	simpld 475	. . . . . . . . . . 11 |- ( ph -> x e. ZZ )
14	13	zcnd 11483	. . . . . . . . . 10 |- ( ph -> x e. CC )
15	11	nnne0d 11065	. . . . . . . . . 10 |- ( ph -> t =/= 0 )
16	2	nnne0d 11065	. . . . . . . . . 10 |- ( ph -> y =/= 0 )
17	6, 12, 14, 3, 15, 16	divmuleqd 10847	. . . . . . . . 9 |- ( ph -> ( ( s / t ) = ( x / y ) <-> ( s x. y ) = ( x x. t ) ) )
18	10, 17	mpbid 222	. . . . . . . 8 |- ( ph -> ( s x. y ) = ( x x. t ) )
19	7, 18	eqtrd 2656	. . . . . . 7 |- ( ph -> ( y x. s ) = ( x x. t ) )
20	19	breq2d 4665	. . . . . 6 |- ( ph -> ( ( P ^ z ) || ( y x. s ) <-> ( P ^ z ) || ( x x. t ) ) )
21	20	rabbidv 3189	. . . . 5 |- ( ph -> { z e. NN0 | ( P ^ z ) || ( y x. s ) } = { z e. NN0 | ( P ^ z ) || ( x x. t ) } )
22		oveq2 6658	. . . . . . 7 |- ( n = z -> ( P ^ n ) = ( P ^ z ) )
23	22	breq1d 4663	. . . . . 6 |- ( n = z -> ( ( P ^ n ) || ( y x. s ) <-> ( P ^ z ) || ( y x. s ) ) )
24	23	cbvrabv 3199	. . . . 5 |- { n e. NN0 | ( P ^ n ) || ( y x. s ) } = { z e. NN0 | ( P ^ z ) || ( y x. s ) }
25	22	breq1d 4663	. . . . . 6 |- ( n = z -> ( ( P ^ n ) || ( x x. t ) <-> ( P ^ z ) || ( x x. t ) ) )
26	25	cbvrabv 3199	. . . . 5 |- { n e. NN0 | ( P ^ n ) || ( x x. t ) } = { z e. NN0 | ( P ^ z ) || ( x x. t ) }
27	21, 24, 26	3eqtr4g 2681	. . . 4 |- ( ph -> { n e. NN0 | ( P ^ n ) || ( y x. s ) } = { n e. NN0 | ( P ^ n ) || ( x x. t ) } )
28	27	supeq1d 8352	. . 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 11481	. . . 4 |- ( ph -> y e. ZZ )
31		pceu.6	. . . . 5 |- ( ph -> N =/= 0 )
32	12, 15	div0d 10800	. . . . . . . 8 |- ( ph -> ( 0 / t ) = 0 )
33		oveq1 6657	. . . . . . . . 9 |- ( s = 0 -> ( s / t ) = ( 0 / t ) )
34	33	eqeq1d 2624	. . . . . . . 8 |- ( s = 0 -> ( ( s / t ) = 0 <-> ( 0 / t ) = 0 ) )
35	32, 34	syl5ibrcom 237	. . . . . . 7 |- ( ph -> ( s = 0 -> ( s / t ) = 0 ) )
36	8	eqeq1d 2624	. . . . . . 7 |- ( ph -> ( N = 0 <-> ( s / t ) = 0 ) )
37	35, 36	sylibrd 249	. . . . . 6 |- ( ph -> ( s = 0 -> N = 0 ) )
38	37	necon3d 2815	. . . . 5 |- ( ph -> ( N =/= 0 -> s =/= 0 ) )
39	31, 38	mpd 15	. . . 4 |- ( ph -> s =/= 0 )
40		pcval.2	. . . . 5 |- T = sup ( { n e. NN0 | ( P ^ n ) || y } , RR , < )
41		pceu.3	. . . . 5 |- U = sup ( { n e. NN0 | ( P ^ n ) || s } , RR , < )
42		eqid 2622	. . . . 5 |- sup ( { n e. NN0 | ( P ^ n ) || ( y x. s ) } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || ( y x. s ) } , RR , < )
43	40, 41, 42	pcpremul 15548	. . . 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 , < ) )
44	29, 30, 16, 5, 39, 43	syl122anc 1335	. . 3 |- ( ph -> ( T + U ) = sup ( { n e. NN0 | ( P ^ n ) || ( y x. s ) } , RR , < ) )
45	3, 16	div0d 10800	. . . . . . . 8 |- ( ph -> ( 0 / y ) = 0 )
46		oveq1 6657	. . . . . . . . 9 |- ( x = 0 -> ( x / y ) = ( 0 / y ) )
47	46	eqeq1d 2624	. . . . . . . 8 |- ( x = 0 -> ( ( x / y ) = 0 <-> ( 0 / y ) = 0 ) )
48	45, 47	syl5ibrcom 237	. . . . . . 7 |- ( ph -> ( x = 0 -> ( x / y ) = 0 ) )
49	9	eqeq1d 2624	. . . . . . 7 |- ( ph -> ( N = 0 <-> ( x / y ) = 0 ) )
50	48, 49	sylibrd 249	. . . . . 6 |- ( ph -> ( x = 0 -> N = 0 ) )
51	50	necon3d 2815	. . . . 5 |- ( ph -> ( N =/= 0 -> x =/= 0 ) )
52	31, 51	mpd 15	. . . 4 |- ( ph -> x =/= 0 )
53	11	nnzd 11481	. . . 4 |- ( ph -> t e. ZZ )
54		pcval.1	. . . . 5 |- S = sup ( { n e. NN0 | ( P ^ n ) || x } , RR , < )
55		pceu.4	. . . . 5 |- V = sup ( { n e. NN0 | ( P ^ n ) || t } , RR , < )
56		eqid 2622	. . . . 5 |- sup ( { n e. NN0 | ( P ^ n ) || ( x x. t ) } , RR , < ) = sup ( { n e. NN0 | ( P ^ n ) || ( x x. t ) } , RR , < )
57	54, 55, 56	pcpremul 15548	. . . 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 , < ) )
58	29, 13, 52, 53, 15, 57	syl122anc 1335	. . 3 |- ( ph -> ( S + V ) = sup ( { n e. NN0 | ( P ^ n ) || ( x x. t ) } , RR , < ) )
59	28, 44, 58	3eqtr4d 2666	. 2 |- ( ph -> ( T + U ) = ( S + V ) )
60		prmuz2 15408	. . . . . 6 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
61	29, 60	syl 17	. . . . 5 |- ( ph -> P e. ( ZZ>= ` 2 ) )
62		eqid 2622	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || y }
63	62, 40	pcprecl 15544	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( y e. ZZ /\ y =/= 0 ) ) -> ( T e. NN0 /\ ( P ^ T ) || y ) )
64	63	simpld 475	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( y e. ZZ /\ y =/= 0 ) ) -> T e. NN0 )
65	61, 30, 16, 64	syl12anc 1324	. . . 4 |- ( ph -> T e. NN0 )
66	65	nn0cnd 11353	. . 3 |- ( ph -> T e. CC )
67		eqid 2622	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || s } = { n e. NN0 | ( P ^ n ) || s }
68	67, 41	pcprecl 15544	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( s e. ZZ /\ s =/= 0 ) ) -> ( U e. NN0 /\ ( P ^ U ) || s ) )
69	68	simpld 475	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( s e. ZZ /\ s =/= 0 ) ) -> U e. NN0 )
70	61, 5, 39, 69	syl12anc 1324	. . . 4 |- ( ph -> U e. NN0 )
71	70	nn0cnd 11353	. . 3 |- ( ph -> U e. CC )
72		eqid 2622	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || x }
73	72, 54	pcprecl 15544	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || x ) )
74	73	simpld 475	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( x e. ZZ /\ x =/= 0 ) ) -> S e. NN0 )
75	61, 13, 52, 74	syl12anc 1324	. . . 4 |- ( ph -> S e. NN0 )
76	75	nn0cnd 11353	. . 3 |- ( ph -> S e. CC )
77		eqid 2622	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || t } = { n e. NN0 | ( P ^ n ) || t }
78	77, 55	pcprecl 15544	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( t e. ZZ /\ t =/= 0 ) ) -> ( V e. NN0 /\ ( P ^ V ) || t ) )
79	78	simpld 475	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( t e. ZZ /\ t =/= 0 ) ) -> V e. NN0 )
80	61, 53, 15, 79	syl12anc 1324	. . . 4 |- ( ph -> V e. NN0 )
81	80	nn0cnd 11353	. . 3 |- ( ph -> V e. CC )
82	66, 71, 76, 81	addsubeq4d 10443	. 2 |- ( ph -> ( ( T + U ) = ( S + V ) <-> ( S - T ) = ( U - V ) ) )
83	59, 82	mpbid 222	1 |- ( ph -> ( S - T ) = ( U - V ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   supcsup 8346   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ^cexp 12860   || cdvds 14983   Primecprime 15385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-prm 15386

This theorem is referenced by:  pceu  15551