Theorem ostthlem2 25317

Description: Lemma for ostth 25328. Refine ostthlem1 25316 so that it is sufficient to only show equality on the primes. (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)

Hypotheses

Ref	Expression
qrng.q	|- Q = (ℂfld ↾s QQ )
qabsabv.a	|- A = (AbsVal ` Q )
ostthlem1.1	|- ( ph -> F e. A )
ostthlem1.2	|- ( ph -> G e. A )
ostthlem2.3	|- ( ( ph /\ p e. Prime ) -> ( F ` p ) = ( G ` p ) )

Assertion

Ref	Expression
ostthlem2	|- ( ph -> F = G )

Distinct variable groups:   G, p   ph, p   A, p   F, p

Allowed substitution hint:   Q( p)

Proof of Theorem ostthlem2

Dummy variables n y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qrng.q	. 2 |- Q = (ℂfld ↾s QQ )
2		qabsabv.a	. 2 |- A = (AbsVal ` Q )
3		ostthlem1.1	. 2 |- ( ph -> F e. A )
4		ostthlem1.2	. 2 |- ( ph -> G e. A )
5		eluz2nn 11726	. . 3 |- ( n e. ( ZZ>= ` 2 ) -> n e. NN )
6		fveq2 6191	. . . . . . 7 |- ( p = 1 -> ( F ` p ) = ( F ` 1 ) )
7		fveq2 6191	. . . . . . 7 |- ( p = 1 -> ( G ` p ) = ( G ` 1 ) )
8	6, 7	eqeq12d 2637	. . . . . 6 |- ( p = 1 -> ( ( F ` p ) = ( G ` p ) <-> ( F ` 1 ) = ( G ` 1 ) ) )
9	8	imbi2d 330	. . . . 5 |- ( p = 1 -> ( ( ph -> ( F ` p ) = ( G ` p ) ) <-> ( ph -> ( F ` 1 ) = ( G ` 1 ) ) ) )
10		fveq2 6191	. . . . . . 7 |- ( p = y -> ( F ` p ) = ( F ` y ) )
11		fveq2 6191	. . . . . . 7 |- ( p = y -> ( G ` p ) = ( G ` y ) )
12	10, 11	eqeq12d 2637	. . . . . 6 |- ( p = y -> ( ( F ` p ) = ( G ` p ) <-> ( F ` y ) = ( G ` y ) ) )
13	12	imbi2d 330	. . . . 5 |- ( p = y -> ( ( ph -> ( F ` p ) = ( G ` p ) ) <-> ( ph -> ( F ` y ) = ( G ` y ) ) ) )
14		fveq2 6191	. . . . . . 7 |- ( p = z -> ( F ` p ) = ( F ` z ) )
15		fveq2 6191	. . . . . . 7 |- ( p = z -> ( G ` p ) = ( G ` z ) )
16	14, 15	eqeq12d 2637	. . . . . 6 |- ( p = z -> ( ( F ` p ) = ( G ` p ) <-> ( F ` z ) = ( G ` z ) ) )
17	16	imbi2d 330	. . . . 5 |- ( p = z -> ( ( ph -> ( F ` p ) = ( G ` p ) ) <-> ( ph -> ( F ` z ) = ( G ` z ) ) ) )
18		fveq2 6191	. . . . . . 7 |- ( p = ( y x. z ) -> ( F ` p ) = ( F ` ( y x. z ) ) )
19		fveq2 6191	. . . . . . 7 |- ( p = ( y x. z ) -> ( G ` p ) = ( G ` ( y x. z ) ) )
20	18, 19	eqeq12d 2637	. . . . . 6 |- ( p = ( y x. z ) -> ( ( F ` p ) = ( G ` p ) <-> ( F ` ( y x. z ) ) = ( G ` ( y x. z ) ) ) )
21	20	imbi2d 330	. . . . 5 |- ( p = ( y x. z ) -> ( ( ph -> ( F ` p ) = ( G ` p ) ) <-> ( ph -> ( F ` ( y x. z ) ) = ( G ` ( y x. z ) ) ) ) )
22		fveq2 6191	. . . . . . 7 |- ( p = n -> ( F ` p ) = ( F ` n ) )
23		fveq2 6191	. . . . . . 7 |- ( p = n -> ( G ` p ) = ( G ` n ) )
24	22, 23	eqeq12d 2637	. . . . . 6 |- ( p = n -> ( ( F ` p ) = ( G ` p ) <-> ( F ` n ) = ( G ` n ) ) )
25	24	imbi2d 330	. . . . 5 |- ( p = n -> ( ( ph -> ( F ` p ) = ( G ` p ) ) <-> ( ph -> ( F ` n ) = ( G ` n ) ) ) )
26		ax-1ne0 10005	. . . . . . 7 |- 1 =/= 0
27	1	qrng1 25311	. . . . . . . 8 |- 1 = ( 1r ` Q )
28	1	qrng0 25310	. . . . . . . 8 |- 0 = ( 0g ` Q )
29	2, 27, 28	abv1z 18832	. . . . . . 7 |- ( ( F e. A /\ 1 =/= 0 ) -> ( F ` 1 ) = 1 )
30	3, 26, 29	sylancl 694	. . . . . 6 |- ( ph -> ( F ` 1 ) = 1 )
31	2, 27, 28	abv1z 18832	. . . . . . 7 |- ( ( G e. A /\ 1 =/= 0 ) -> ( G ` 1 ) = 1 )
32	4, 26, 31	sylancl 694	. . . . . 6 |- ( ph -> ( G ` 1 ) = 1 )
33	30, 32	eqtr4d 2659	. . . . 5 |- ( ph -> ( F ` 1 ) = ( G ` 1 ) )
34		ostthlem2.3	. . . . . 6 |- ( ( ph /\ p e. Prime ) -> ( F ` p ) = ( G ` p ) )
35	34	expcom 451	. . . . 5 |- ( p e. Prime -> ( ph -> ( F ` p ) = ( G ` p ) ) )
36		jcab 907	. . . . . 6 |- ( ( ph -> ( ( F ` y ) = ( G ` y ) /\ ( F ` z ) = ( G ` z ) ) ) <-> ( ( ph -> ( F ` y ) = ( G ` y ) ) /\ ( ph -> ( F ` z ) = ( G ` z ) ) ) )
37		oveq12 6659	. . . . . . . . 9 |- ( ( ( F ` y ) = ( G ` y ) /\ ( F ` z ) = ( G ` z ) ) -> ( ( F ` y ) x. ( F ` z ) ) = ( ( G ` y ) x. ( G ` z ) ) )
38	3	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) ) -> F e. A )
39		eluzelz 11697	. . . . . . . . . . . . 13 |- ( y e. ( ZZ>= ` 2 ) -> y e. ZZ )
40	39	ad2antrl 764	. . . . . . . . . . . 12 |- ( ( ph /\ ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) ) -> y e. ZZ )
41		zq 11794	. . . . . . . . . . . 12 |- ( y e. ZZ -> y e. QQ )
42	40, 41	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) ) -> y e. QQ )
43		eluzelz 11697	. . . . . . . . . . . . 13 |- ( z e. ( ZZ>= ` 2 ) -> z e. ZZ )
44	43	ad2antll 765	. . . . . . . . . . . 12 |- ( ( ph /\ ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) ) -> z e. ZZ )
45		zq 11794	. . . . . . . . . . . 12 |- ( z e. ZZ -> z e. QQ )
46	44, 45	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) ) -> z e. QQ )
47	1	qrngbas 25308	. . . . . . . . . . . 12 |- QQ = ( Base ` Q )
48		qex 11800	. . . . . . . . . . . . 13 |- QQ e. _V
49		cnfldmul 19752	. . . . . . . . . . . . . 14 |- x. = ( .r ` ℂfld )
50	1, 49	ressmulr 16006	. . . . . . . . . . . . 13 |- ( QQ e. _V -> x. = ( .r ` Q ) )
51	48, 50	ax-mp 5	. . . . . . . . . . . 12 |- x. = ( .r ` Q )
52	2, 47, 51	abvmul 18829	. . . . . . . . . . 11 |- ( ( F e. A /\ y e. QQ /\ z e. QQ ) -> ( F ` ( y x. z ) ) = ( ( F ` y ) x. ( F ` z ) ) )
53	38, 42, 46, 52	syl3anc 1326	. . . . . . . . . 10 |- ( ( ph /\ ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) ) -> ( F ` ( y x. z ) ) = ( ( F ` y ) x. ( F ` z ) ) )
54	4	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) ) -> G e. A )
55	2, 47, 51	abvmul 18829	. . . . . . . . . . 11 |- ( ( G e. A /\ y e. QQ /\ z e. QQ ) -> ( G ` ( y x. z ) ) = ( ( G ` y ) x. ( G ` z ) ) )
56	54, 42, 46, 55	syl3anc 1326	. . . . . . . . . 10 |- ( ( ph /\ ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) ) -> ( G ` ( y x. z ) ) = ( ( G ` y ) x. ( G ` z ) ) )
57	53, 56	eqeq12d 2637	. . . . . . . . 9 |- ( ( ph /\ ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) ) -> ( ( F ` ( y x. z ) ) = ( G ` ( y x. z ) ) <-> ( ( F ` y ) x. ( F ` z ) ) = ( ( G ` y ) x. ( G ` z ) ) ) )
58	37, 57	syl5ibr 236	. . . . . . . 8 |- ( ( ph /\ ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) ) -> ( ( ( F ` y ) = ( G ` y ) /\ ( F ` z ) = ( G ` z ) ) -> ( F ` ( y x. z ) ) = ( G ` ( y x. z ) ) ) )
59	58	expcom 451	. . . . . . 7 |- ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) -> ( ph -> ( ( ( F ` y ) = ( G ` y ) /\ ( F ` z ) = ( G ` z ) ) -> ( F ` ( y x. z ) ) = ( G ` ( y x. z ) ) ) ) )
60	59	a2d 29	. . . . . 6 |- ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) -> ( ( ph -> ( ( F ` y ) = ( G ` y ) /\ ( F ` z ) = ( G ` z ) ) ) -> ( ph -> ( F ` ( y x. z ) ) = ( G ` ( y x. z ) ) ) ) )
61	36, 60	syl5bir 233	. . . . 5 |- ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) -> ( ( ( ph -> ( F ` y ) = ( G ` y ) ) /\ ( ph -> ( F ` z ) = ( G ` z ) ) ) -> ( ph -> ( F ` ( y x. z ) ) = ( G ` ( y x. z ) ) ) ) )
62	9, 13, 17, 21, 25, 33, 35, 61	prmind 15399	. . . 4 |- ( n e. NN -> ( ph -> ( F ` n ) = ( G ` n ) ) )
63	62	impcom 446	. . 3 |- ( ( ph /\ n e. NN ) -> ( F ` n ) = ( G ` n ) )
64	5, 63	sylan2 491	. 2 |- ( ( ph /\ n e. ( ZZ>= ` 2 ) ) -> ( F ` n ) = ( G ` n ) )
65	1, 2, 3, 4, 64	ostthlem1 25316	1 |- ( ph -> F = G )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   x. cmul 9941   NNcn 11020   2c2 11070   ZZcz 11377   ZZ>=cuz 11687   QQcq 11788   Primecprime 15385   ↾_s cress 15858   .rcmulr 15942  AbsValcabv 18816  ℂ_fldccnfld 19746

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-rep 4771  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  ax-addf 10015  ax-mulf 10016

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-int 4476  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-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-ico 12181  df-fz 12327  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-prm 15386  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591  df-cmn 18195  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-subrg 18778  df-abv 18817  df-cnfld 19747

This theorem is referenced by:  ostth1  25322  ostth3  25327