Theorem pceulem 12992

Description: Lemma for pceu 12993. (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 9251	. . . . . . . . 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 9701	. . . . . . . . 9 |- ( ph -> s e. CC )
7	3, 6	mulcomd 8295	. . . . . . . 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 2267	. . . . . . . . 9 |- ( ph -> ( s / t ) = ( x / y ) )
11	4	simprd 114	. . . . . . . . . . 11 |- ( ph -> t e. NN )
12	11	nncnd 9251	. . . . . . . . . 10 |- ( ph -> t e. CC )
13	1	simpld 112	. . . . . . . . . . 11 |- ( ph -> x e. ZZ )
14	13	zcnd 9701	. . . . . . . . . 10 |- ( ph -> x e. CC )
15	11	nnap0d 9283	. . . . . . . . . 10 |- ( ph -> t # 0 )
16	2	nnap0d 9283	. . . . . . . . . 10 |- ( ph -> y # 0 )
17	6, 12, 14, 3, 15, 16	divmuleqapd 9107	. . . . . . . . 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 2265	. . . . . . 7 |- ( ph -> ( y x. s ) = ( x x. t ) )
20	19	breq2d 4121	. . . . . 6 |- ( ph -> ( ( P ^ z ) || ( y x. s ) <-> ( P ^ z ) || ( x x. t ) ) )
21	20	rabbidv 2802	. . . . 5 |- ( ph -> { z e. NN0 | ( P ^ z ) || ( y x. s ) } = { z e. NN0 | ( P ^ z ) || ( x x. t ) } )
22		oveq2 6058	. . . . . . 7 |- ( n = z -> ( P ^ n ) = ( P ^ z ) )
23	22	breq1d 4119	. . . . . 6 |- ( n = z -> ( ( P ^ n ) || ( y x. s ) <-> ( P ^ z ) || ( y x. s ) ) )
24	23	cbvrabv 2812	. . . . 5 |- { n e. NN0 | ( P ^ n ) || ( y x. s ) } = { z e. NN0 | ( P ^ z ) || ( y x. s ) }
25	22	breq1d 4119	. . . . . 6 |- ( n = z -> ( ( P ^ n ) || ( x x. t ) <-> ( P ^ z ) || ( x x. t ) ) )
26	25	cbvrabv 2812	. . . . 5 |- { n e. NN0 | ( P ^ n ) || ( x x. t ) } = { z e. NN0 | ( P ^ z ) || ( x x. t ) }
27	21, 24, 26	3eqtr4g 2290	. . . 4 |- ( ph -> { n e. NN0 | ( P ^ n ) || ( y x. s ) } = { n e. NN0 | ( P ^ n ) || ( x x. t ) } )
28	27	supeq1d 7278	. . 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 9699	. . . 4 |- ( ph -> y e. ZZ )
31	2	nnne0d 9282	. . . 4 |- ( ph -> y =/= 0 )
32		pceu.6	. . . . 5 |- ( ph -> N =/= 0 )
33	12, 15	div0apd 9061	. . . . . . . 8 |- ( ph -> ( 0 / t ) = 0 )
34		oveq1 6057	. . . . . . . . 9 |- ( s = 0 -> ( s / t ) = ( 0 / t ) )
35	34	eqeq1d 2241	. . . . . . . 8 |- ( s = 0 -> ( ( s / t ) = 0 <-> ( 0 / t ) = 0 ) )
36	33, 35	syl5ibrcom 157	. . . . . . 7 |- ( ph -> ( s = 0 -> ( s / t ) = 0 ) )
37	8	eqeq1d 2241	. . . . . . 7 |- ( ph -> ( N = 0 <-> ( s / t ) = 0 ) )
38	36, 37	sylibrd 169	. . . . . 6 |- ( ph -> ( s = 0 -> N = 0 ) )
39	38	necon3d 2456	. . . . 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 2232	. . . . 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 12991	. . . 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 1283	. . 3 |- ( ph -> ( T + U ) = sup ( { n e. NN0 | ( P ^ n ) || ( y x. s ) } , RR , < ) )
46	3, 16	div0apd 9061	. . . . . . . 8 |- ( ph -> ( 0 / y ) = 0 )
47		oveq1 6057	. . . . . . . . 9 |- ( x = 0 -> ( x / y ) = ( 0 / y ) )
48	47	eqeq1d 2241	. . . . . . . 8 |- ( x = 0 -> ( ( x / y ) = 0 <-> ( 0 / y ) = 0 ) )
49	46, 48	syl5ibrcom 157	. . . . . . 7 |- ( ph -> ( x = 0 -> ( x / y ) = 0 ) )
50	9	eqeq1d 2241	. . . . . . 7 |- ( ph -> ( N = 0 <-> ( x / y ) = 0 ) )
51	49, 50	sylibrd 169	. . . . . 6 |- ( ph -> ( x = 0 -> N = 0 ) )
52	51	necon3d 2456	. . . . 5 |- ( ph -> ( N =/= 0 -> x =/= 0 ) )
53	32, 52	mpd 13	. . . 4 |- ( ph -> x =/= 0 )
54	11	nnzd 9699	. . . 4 |- ( ph -> t e. ZZ )
55	11	nnne0d 9282	. . . 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 2232	. . . . 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 12991	. . . 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 1283	. . 3 |- ( ph -> ( S + V ) = sup ( { n e. NN0 | ( P ^ n ) || ( x x. t ) } , RR , < ) )
61	28, 45, 60	3eqtr4d 2275	. 2 |- ( ph -> ( T + U ) = ( S + V ) )
62		prmuz2 12828	. . . . . 6 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
63	29, 62	syl 14	. . . . 5 |- ( ph -> P e. ( ZZ>= ` 2 ) )
64		eqid 2232	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || y }
65	64, 41	pcprecl 12987	. . . . . 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 1272	. . . 4 |- ( ph -> T e. NN0 )
68	67	nn0cnd 9555	. . 3 |- ( ph -> T e. CC )
69		eqid 2232	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || s } = { n e. NN0 | ( P ^ n ) || s }
70	69, 42	pcprecl 12987	. . . . . 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 1272	. . . 4 |- ( ph -> U e. NN0 )
73	72	nn0cnd 9555	. . 3 |- ( ph -> U e. CC )
74		eqid 2232	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || x }
75	74, 56	pcprecl 12987	. . . . . 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 1272	. . . 4 |- ( ph -> S e. NN0 )
78	77	nn0cnd 9555	. . 3 |- ( ph -> S e. CC )
79		eqid 2232	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || t } = { n e. NN0 | ( P ^ n ) || t }
80	79, 57	pcprecl 12987	. . . . . 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 1272	. . . 4 |- ( ph -> V e. NN0 )
83	82	nn0cnd 9555	. . 3 |- ( ph -> V e. CC )
84	68, 73, 78, 83	addsubeq4d 8635	. 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 1398   e. wcel 2203   =/= wne 2412   {crab 2524   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   supcsup 7273   RRcr 8126   0cc0 8127   + caddc 8130   x. cmul 8132   < clt 8308   - cmin 8444   / cdiv 8946   NNcn 9237   2c2 9288   NN0cn0 9496   ZZcz 9577   ZZ>=cuz 9853   ^cexp 10900   || cdvds 12473   Primecprime 12804

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-2o 6648  df-er 6767  df-en 6976  df-sup 7275  df-inf 7276  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-dvds 12474  df-gcd 12650  df-prm 12805

This theorem is referenced by:  pceu  12993