Theorem pceulem 12235

Description: Lemma for pceu 12236. (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 113	. . . . . . . . . 10 |- ( ph -> y e. NN )
3	2	nncnd 8879	. . . . . . . . 9 |- ( ph -> y e. CC )
4		pceu.9	. . . . . . . . . . 11 |- ( ph -> ( s e. ZZ /\ t e. NN ) )
5	4	simpld 111	. . . . . . . . . 10 |- ( ph -> s e. ZZ )
6	5	zcnd 9322	. . . . . . . . 9 |- ( ph -> s e. CC )
7	3, 6	mulcomd 7928	. . . . . . . 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 2205	. . . . . . . . 9 |- ( ph -> ( s / t ) = ( x / y ) )
11	4	simprd 113	. . . . . . . . . . 11 |- ( ph -> t e. NN )
12	11	nncnd 8879	. . . . . . . . . 10 |- ( ph -> t e. CC )
13	1	simpld 111	. . . . . . . . . . 11 |- ( ph -> x e. ZZ )
14	13	zcnd 9322	. . . . . . . . . 10 |- ( ph -> x e. CC )
15	11	nnap0d 8911	. . . . . . . . . 10 |- ( ph -> t # 0 )
16	2	nnap0d 8911	. . . . . . . . . 10 |- ( ph -> y # 0 )
17	6, 12, 14, 3, 15, 16	divmuleqapd 8737	. . . . . . . . 9 |- ( ph -> ( ( s / t ) = ( x / y ) <-> ( s x. y ) = ( x x. t ) ) )
18	10, 17	mpbid 146	. . . . . . . 8 |- ( ph -> ( s x. y ) = ( x x. t ) )
19	7, 18	eqtrd 2203	. . . . . . 7 |- ( ph -> ( y x. s ) = ( x x. t ) )
20	19	breq2d 3999	. . . . . 6 |- ( ph -> ( ( P ^ z ) || ( y x. s ) <-> ( P ^ z ) || ( x x. t ) ) )
21	20	rabbidv 2719	. . . . 5 |- ( ph -> { z e. NN0 | ( P ^ z ) || ( y x. s ) } = { z e. NN0 | ( P ^ z ) || ( x x. t ) } )
22		oveq2 5858	. . . . . . 7 |- ( n = z -> ( P ^ n ) = ( P ^ z ) )
23	22	breq1d 3997	. . . . . 6 |- ( n = z -> ( ( P ^ n ) || ( y x. s ) <-> ( P ^ z ) || ( y x. s ) ) )
24	23	cbvrabv 2729	. . . . 5 |- { n e. NN0 | ( P ^ n ) || ( y x. s ) } = { z e. NN0 | ( P ^ z ) || ( y x. s ) }
25	22	breq1d 3997	. . . . . 6 |- ( n = z -> ( ( P ^ n ) || ( x x. t ) <-> ( P ^ z ) || ( x x. t ) ) )
26	25	cbvrabv 2729	. . . . 5 |- { n e. NN0 | ( P ^ n ) || ( x x. t ) } = { z e. NN0 | ( P ^ z ) || ( x x. t ) }
27	21, 24, 26	3eqtr4g 2228	. . . 4 |- ( ph -> { n e. NN0 | ( P ^ n ) || ( y x. s ) } = { n e. NN0 | ( P ^ n ) || ( x x. t ) } )
28	27	supeq1d 6960	. . 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 9320	. . . 4 |- ( ph -> y e. ZZ )
31	2	nnne0d 8910	. . . 4 |- ( ph -> y =/= 0 )
32		pceu.6	. . . . 5 |- ( ph -> N =/= 0 )
33	12, 15	div0apd 8691	. . . . . . . 8 |- ( ph -> ( 0 / t ) = 0 )
34		oveq1 5857	. . . . . . . . 9 |- ( s = 0 -> ( s / t ) = ( 0 / t ) )
35	34	eqeq1d 2179	. . . . . . . 8 |- ( s = 0 -> ( ( s / t ) = 0 <-> ( 0 / t ) = 0 ) )
36	33, 35	syl5ibrcom 156	. . . . . . 7 |- ( ph -> ( s = 0 -> ( s / t ) = 0 ) )
37	8	eqeq1d 2179	. . . . . . 7 |- ( ph -> ( N = 0 <-> ( s / t ) = 0 ) )
38	36, 37	sylibrd 168	. . . . . 6 |- ( ph -> ( s = 0 -> N = 0 ) )
39	38	necon3d 2384	. . . . 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 2170	. . . . 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 12234	. . . 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 1242	. . 3 |- ( ph -> ( T + U ) = sup ( { n e. NN0 | ( P ^ n ) || ( y x. s ) } , RR , < ) )
46	3, 16	div0apd 8691	. . . . . . . 8 |- ( ph -> ( 0 / y ) = 0 )
47		oveq1 5857	. . . . . . . . 9 |- ( x = 0 -> ( x / y ) = ( 0 / y ) )
48	47	eqeq1d 2179	. . . . . . . 8 |- ( x = 0 -> ( ( x / y ) = 0 <-> ( 0 / y ) = 0 ) )
49	46, 48	syl5ibrcom 156	. . . . . . 7 |- ( ph -> ( x = 0 -> ( x / y ) = 0 ) )
50	9	eqeq1d 2179	. . . . . . 7 |- ( ph -> ( N = 0 <-> ( x / y ) = 0 ) )
51	49, 50	sylibrd 168	. . . . . 6 |- ( ph -> ( x = 0 -> N = 0 ) )
52	51	necon3d 2384	. . . . 5 |- ( ph -> ( N =/= 0 -> x =/= 0 ) )
53	32, 52	mpd 13	. . . 4 |- ( ph -> x =/= 0 )
54	11	nnzd 9320	. . . 4 |- ( ph -> t e. ZZ )
55	11	nnne0d 8910	. . . 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 2170	. . . . 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 12234	. . . 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 1242	. . 3 |- ( ph -> ( S + V ) = sup ( { n e. NN0 | ( P ^ n ) || ( x x. t ) } , RR , < ) )
61	28, 45, 60	3eqtr4d 2213	. 2 |- ( ph -> ( T + U ) = ( S + V ) )
62		prmuz2 12072	. . . . . 6 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
63	29, 62	syl 14	. . . . 5 |- ( ph -> P e. ( ZZ>= ` 2 ) )
64		eqid 2170	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || y }
65	64, 41	pcprecl 12230	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( y e. ZZ /\ y =/= 0 ) ) -> ( T e. NN0 /\ ( P ^ T ) || y ) )
66	65	simpld 111	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( y e. ZZ /\ y =/= 0 ) ) -> T e. NN0 )
67	63, 30, 31, 66	syl12anc 1231	. . . 4 |- ( ph -> T e. NN0 )
68	67	nn0cnd 9177	. . 3 |- ( ph -> T e. CC )
69		eqid 2170	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || s } = { n e. NN0 | ( P ^ n ) || s }
70	69, 42	pcprecl 12230	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( s e. ZZ /\ s =/= 0 ) ) -> ( U e. NN0 /\ ( P ^ U ) || s ) )
71	70	simpld 111	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( s e. ZZ /\ s =/= 0 ) ) -> U e. NN0 )
72	63, 5, 40, 71	syl12anc 1231	. . . 4 |- ( ph -> U e. NN0 )
73	72	nn0cnd 9177	. . 3 |- ( ph -> U e. CC )
74		eqid 2170	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || x }
75	74, 56	pcprecl 12230	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( x e. ZZ /\ x =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || x ) )
76	75	simpld 111	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( x e. ZZ /\ x =/= 0 ) ) -> S e. NN0 )
77	63, 13, 53, 76	syl12anc 1231	. . . 4 |- ( ph -> S e. NN0 )
78	77	nn0cnd 9177	. . 3 |- ( ph -> S e. CC )
79		eqid 2170	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || t } = { n e. NN0 | ( P ^ n ) || t }
80	79, 57	pcprecl 12230	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( t e. ZZ /\ t =/= 0 ) ) -> ( V e. NN0 /\ ( P ^ V ) || t ) )
81	80	simpld 111	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( t e. ZZ /\ t =/= 0 ) ) -> V e. NN0 )
82	63, 54, 55, 81	syl12anc 1231	. . . 4 |- ( ph -> V e. NN0 )
83	82	nn0cnd 9177	. . 3 |- ( ph -> V e. CC )
84	68, 73, 78, 83	addsubeq4d 8268	. 2 |- ( ph -> ( ( T + U ) = ( S + V ) <-> ( S - T ) = ( U - V ) ) )
85	61, 84	mpbid 146	1 |- ( ph -> ( S - T ) = ( U - V ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   =/= wne 2340   {crab 2452   class class class wbr 3987   ` cfv 5196  (class class class)co 5850   supcsup 6955   RRcr 7760   0cc0 7761   + caddc 7764   x. cmul 7766   < clt 7941   - cmin 8077   / cdiv 8576   NNcn 8865   2c2 8916   NN0cn0 9122   ZZcz 9199   ZZ>=cuz 9474   ^cexp 10462   || cdvds 11736   Primecprime 12048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879  ax-arch 7880  ax-caucvg 7881

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3526  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-po 4279  df-iso 4280  df-iord 4349  df-on 4351  df-ilim 4352  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-isom 5205  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-recs 6281  df-frec 6367  df-1o 6392  df-2o 6393  df-er 6509  df-en 6715  df-sup 6957  df-inf 6958  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488  df-div 8577  df-inn 8866  df-2 8924  df-3 8925  df-4 8926  df-n0 9123  df-z 9200  df-uz 9475  df-q 9566  df-rp 9598  df-fz 9953  df-fzo 10086  df-fl 10213  df-mod 10266  df-seqfrec 10389  df-exp 10463  df-cj 10793  df-re 10794  df-im 10795  df-rsqrt 10949  df-abs 10950  df-dvds 11737  df-gcd 11885  df-prm 12049

This theorem is referenced by:  pceu  12236