Theorem pceulem 12617

Description: Lemma for pceu 12618. (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 9050	. . . . . . . . 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 9496	. . . . . . . . 9 |- ( ph -> s e. CC )
7	3, 6	mulcomd 8094	. . . . . . . 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 2240	. . . . . . . . 9 |- ( ph -> ( s / t ) = ( x / y ) )
11	4	simprd 114	. . . . . . . . . . 11 |- ( ph -> t e. NN )
12	11	nncnd 9050	. . . . . . . . . 10 |- ( ph -> t e. CC )
13	1	simpld 112	. . . . . . . . . . 11 |- ( ph -> x e. ZZ )
14	13	zcnd 9496	. . . . . . . . . 10 |- ( ph -> x e. CC )
15	11	nnap0d 9082	. . . . . . . . . 10 |- ( ph -> t # 0 )
16	2	nnap0d 9082	. . . . . . . . . 10 |- ( ph -> y # 0 )
17	6, 12, 14, 3, 15, 16	divmuleqapd 8906	. . . . . . . . 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 2238	. . . . . . 7 |- ( ph -> ( y x. s ) = ( x x. t ) )
20	19	breq2d 4056	. . . . . 6 |- ( ph -> ( ( P ^ z ) || ( y x. s ) <-> ( P ^ z ) || ( x x. t ) ) )
21	20	rabbidv 2761	. . . . 5 |- ( ph -> { z e. NN0 | ( P ^ z ) || ( y x. s ) } = { z e. NN0 | ( P ^ z ) || ( x x. t ) } )
22		oveq2 5952	. . . . . . 7 |- ( n = z -> ( P ^ n ) = ( P ^ z ) )
23	22	breq1d 4054	. . . . . 6 |- ( n = z -> ( ( P ^ n ) || ( y x. s ) <-> ( P ^ z ) || ( y x. s ) ) )
24	23	cbvrabv 2771	. . . . 5 |- { n e. NN0 | ( P ^ n ) || ( y x. s ) } = { z e. NN0 | ( P ^ z ) || ( y x. s ) }
25	22	breq1d 4054	. . . . . 6 |- ( n = z -> ( ( P ^ n ) || ( x x. t ) <-> ( P ^ z ) || ( x x. t ) ) )
26	25	cbvrabv 2771	. . . . 5 |- { n e. NN0 | ( P ^ n ) || ( x x. t ) } = { z e. NN0 | ( P ^ z ) || ( x x. t ) }
27	21, 24, 26	3eqtr4g 2263	. . . 4 |- ( ph -> { n e. NN0 | ( P ^ n ) || ( y x. s ) } = { n e. NN0 | ( P ^ n ) || ( x x. t ) } )
28	27	supeq1d 7089	. . 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 9494	. . . 4 |- ( ph -> y e. ZZ )
31	2	nnne0d 9081	. . . 4 |- ( ph -> y =/= 0 )
32		pceu.6	. . . . 5 |- ( ph -> N =/= 0 )
33	12, 15	div0apd 8860	. . . . . . . 8 |- ( ph -> ( 0 / t ) = 0 )
34		oveq1 5951	. . . . . . . . 9 |- ( s = 0 -> ( s / t ) = ( 0 / t ) )
35	34	eqeq1d 2214	. . . . . . . 8 |- ( s = 0 -> ( ( s / t ) = 0 <-> ( 0 / t ) = 0 ) )
36	33, 35	syl5ibrcom 157	. . . . . . 7 |- ( ph -> ( s = 0 -> ( s / t ) = 0 ) )
37	8	eqeq1d 2214	. . . . . . 7 |- ( ph -> ( N = 0 <-> ( s / t ) = 0 ) )
38	36, 37	sylibrd 169	. . . . . 6 |- ( ph -> ( s = 0 -> N = 0 ) )
39	38	necon3d 2420	. . . . 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 2205	. . . . 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 12616	. . . 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 1259	. . 3 |- ( ph -> ( T + U ) = sup ( { n e. NN0 | ( P ^ n ) || ( y x. s ) } , RR , < ) )
46	3, 16	div0apd 8860	. . . . . . . 8 |- ( ph -> ( 0 / y ) = 0 )
47		oveq1 5951	. . . . . . . . 9 |- ( x = 0 -> ( x / y ) = ( 0 / y ) )
48	47	eqeq1d 2214	. . . . . . . 8 |- ( x = 0 -> ( ( x / y ) = 0 <-> ( 0 / y ) = 0 ) )
49	46, 48	syl5ibrcom 157	. . . . . . 7 |- ( ph -> ( x = 0 -> ( x / y ) = 0 ) )
50	9	eqeq1d 2214	. . . . . . 7 |- ( ph -> ( N = 0 <-> ( x / y ) = 0 ) )
51	49, 50	sylibrd 169	. . . . . 6 |- ( ph -> ( x = 0 -> N = 0 ) )
52	51	necon3d 2420	. . . . 5 |- ( ph -> ( N =/= 0 -> x =/= 0 ) )
53	32, 52	mpd 13	. . . 4 |- ( ph -> x =/= 0 )
54	11	nnzd 9494	. . . 4 |- ( ph -> t e. ZZ )
55	11	nnne0d 9081	. . . 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 2205	. . . . 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 12616	. . . 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 1259	. . 3 |- ( ph -> ( S + V ) = sup ( { n e. NN0 | ( P ^ n ) || ( x x. t ) } , RR , < ) )
61	28, 45, 60	3eqtr4d 2248	. 2 |- ( ph -> ( T + U ) = ( S + V ) )
62		prmuz2 12453	. . . . . 6 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
63	29, 62	syl 14	. . . . 5 |- ( ph -> P e. ( ZZ>= ` 2 ) )
64		eqid 2205	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || y } = { n e. NN0 | ( P ^ n ) || y }
65	64, 41	pcprecl 12612	. . . . . 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 1248	. . . 4 |- ( ph -> T e. NN0 )
68	67	nn0cnd 9350	. . 3 |- ( ph -> T e. CC )
69		eqid 2205	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || s } = { n e. NN0 | ( P ^ n ) || s }
70	69, 42	pcprecl 12612	. . . . . 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 1248	. . . 4 |- ( ph -> U e. NN0 )
73	72	nn0cnd 9350	. . 3 |- ( ph -> U e. CC )
74		eqid 2205	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || x } = { n e. NN0 | ( P ^ n ) || x }
75	74, 56	pcprecl 12612	. . . . . 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 1248	. . . 4 |- ( ph -> S e. NN0 )
78	77	nn0cnd 9350	. . 3 |- ( ph -> S e. CC )
79		eqid 2205	. . . . . . 7 |- { n e. NN0 | ( P ^ n ) || t } = { n e. NN0 | ( P ^ n ) || t }
80	79, 57	pcprecl 12612	. . . . . 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 1248	. . . 4 |- ( ph -> V e. NN0 )
83	82	nn0cnd 9350	. . 3 |- ( ph -> V e. CC )
84	68, 73, 78, 83	addsubeq4d 8434	. 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 1373   e. wcel 2176   =/= wne 2376   {crab 2488   class class class wbr 4044   ` cfv 5271  (class class class)co 5944   supcsup 7084   RRcr 7924   0cc0 7925   + caddc 7928   x. cmul 7930   < clt 8107   - cmin 8243   / cdiv 8745   NNcn 9036   2c2 9087   NN0cn0 9295   ZZcz 9372   ZZ>=cuz 9648   ^cexp 10683   || cdvds 12098   Primecprime 12429

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043  ax-arch 8044  ax-caucvg 8045

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-1o 6502  df-2o 6503  df-er 6620  df-en 6828  df-sup 7086  df-inf 7087  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-n0 9296  df-z 9373  df-uz 9649  df-q 9741  df-rp 9776  df-fz 10131  df-fzo 10265  df-fl 10413  df-mod 10468  df-seqfrec 10593  df-exp 10684  df-cj 11153  df-re 11154  df-im 11155  df-rsqrt 11309  df-abs 11310  df-dvds 12099  df-gcd 12275  df-prm 12430

This theorem is referenced by:  pceu  12618