Theorem prmdvdsfmtnof1lem2 41497

Description: Lemma 2 for prmdvdsfmtnof1 41499. (Contributed by AV, 3-Aug-2021.)

Assertion

Ref	Expression
prmdvdsfmtnof1lem2	|- ( ( F e. ran FermatNo /\ G e. ran FermatNo ) -> ( ( I e. Prime /\ I || F /\ I || G ) -> F = G ) )

Proof of Theorem prmdvdsfmtnof1lem2

Dummy variables m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmtnorn 41446	. 2 |- ( F e. ran FermatNo <-> E. n e. NN0 (FermatNo ` n ) = F )
2		fmtnorn 41446	. 2 |- ( G e. ran FermatNo <-> E. m e. NN0 (FermatNo ` m ) = G )
3		2a1 28	. . . . . . . 8 |- ( F = G -> ( (FermatNo ` n ) = F -> ( ( I e. Prime /\ I || F /\ I || G ) -> F = G ) ) )
4	3	2a1d 26	. . . . . . 7 |- ( F = G -> ( ( n e. NN0 /\ m e. NN0 ) -> ( (FermatNo ` m ) = G -> ( (FermatNo ` n ) = F -> ( ( I e. Prime /\ I || F /\ I || G ) -> F = G ) ) ) ) )
5		fmtnonn 41443	. . . . . . . . . . . 12 |- ( n e. NN0 -> (FermatNo ` n ) e. NN )
6	5	ad2antrl 764	. . . . . . . . . . 11 |- ( ( -. F = G /\ ( n e. NN0 /\ m e. NN0 ) ) -> (FermatNo ` n ) e. NN )
7	6	adantr 481	. . . . . . . . . 10 |- ( ( ( -. F = G /\ ( n e. NN0 /\ m e. NN0 ) ) /\ ( (FermatNo ` m ) = G /\ (FermatNo ` n ) = F ) ) -> (FermatNo ` n ) e. NN )
8		eleq1 2689	. . . . . . . . . . 11 |- ( (FermatNo ` n ) = F -> ( (FermatNo ` n ) e. NN <-> F e. NN ) )
9	8	ad2antll 765	. . . . . . . . . 10 |- ( ( ( -. F = G /\ ( n e. NN0 /\ m e. NN0 ) ) /\ ( (FermatNo ` m ) = G /\ (FermatNo ` n ) = F ) ) -> ( (FermatNo ` n ) e. NN <-> F e. NN ) )
10	7, 9	mpbid 222	. . . . . . . . 9 |- ( ( ( -. F = G /\ ( n e. NN0 /\ m e. NN0 ) ) /\ ( (FermatNo ` m ) = G /\ (FermatNo ` n ) = F ) ) -> F e. NN )
11		fmtnonn 41443	. . . . . . . . . . . 12 |- ( m e. NN0 -> (FermatNo ` m ) e. NN )
12	11	ad2antll 765	. . . . . . . . . . 11 |- ( ( -. F = G /\ ( n e. NN0 /\ m e. NN0 ) ) -> (FermatNo ` m ) e. NN )
13	12	adantr 481	. . . . . . . . . 10 |- ( ( ( -. F = G /\ ( n e. NN0 /\ m e. NN0 ) ) /\ ( (FermatNo ` m ) = G /\ (FermatNo ` n ) = F ) ) -> (FermatNo ` m ) e. NN )
14		eleq1 2689	. . . . . . . . . . 11 |- ( (FermatNo ` m ) = G -> ( (FermatNo ` m ) e. NN <-> G e. NN ) )
15	14	ad2antrl 764	. . . . . . . . . 10 |- ( ( ( -. F = G /\ ( n e. NN0 /\ m e. NN0 ) ) /\ ( (FermatNo ` m ) = G /\ (FermatNo ` n ) = F ) ) -> ( (FermatNo ` m ) e. NN <-> G e. NN ) )
16	13, 15	mpbid 222	. . . . . . . . 9 |- ( ( ( -. F = G /\ ( n e. NN0 /\ m e. NN0 ) ) /\ ( (FermatNo ` m ) = G /\ (FermatNo ` n ) = F ) ) -> G e. NN )
17		simpll 790	. . . . . . . . . . . . . . 15 |- ( ( ( n e. NN0 /\ m e. NN0 ) /\ -. (FermatNo ` n ) = (FermatNo ` m ) ) -> n e. NN0 )
18		simplr 792	. . . . . . . . . . . . . . 15 |- ( ( ( n e. NN0 /\ m e. NN0 ) /\ -. (FermatNo ` n ) = (FermatNo ` m ) ) -> m e. NN0 )
19		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( n = m -> (FermatNo ` n ) = (FermatNo ` m ) )
20	19	con3i 150	. . . . . . . . . . . . . . . . 17 |- ( -. (FermatNo ` n ) = (FermatNo ` m ) -> -. n = m )
21	20	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( n e. NN0 /\ m e. NN0 ) /\ -. (FermatNo ` n ) = (FermatNo ` m ) ) -> -. n = m )
22	21	neqned 2801	. . . . . . . . . . . . . . 15 |- ( ( ( n e. NN0 /\ m e. NN0 ) /\ -. (FermatNo ` n ) = (FermatNo ` m ) ) -> n =/= m )
23		goldbachth 41459	. . . . . . . . . . . . . . 15 |- ( ( n e. NN0 /\ m e. NN0 /\ n =/= m ) -> ( (FermatNo ` n ) gcd (FermatNo ` m ) ) = 1 )
24	17, 18, 22, 23	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( ( n e. NN0 /\ m e. NN0 ) /\ -. (FermatNo ` n ) = (FermatNo ` m ) ) -> ( (FermatNo ` n ) gcd (FermatNo ` m ) ) = 1 )
25	24	ex 450	. . . . . . . . . . . . 13 |- ( ( n e. NN0 /\ m e. NN0 ) -> ( -. (FermatNo ` n ) = (FermatNo ` m ) -> ( (FermatNo ` n ) gcd (FermatNo ` m ) ) = 1 ) )
26		eqeq12 2635	. . . . . . . . . . . . . . . 16 |- ( ( (FermatNo ` n ) = F /\ (FermatNo ` m ) = G ) -> ( (FermatNo ` n ) = (FermatNo ` m ) <-> F = G ) )
27	26	notbid 308	. . . . . . . . . . . . . . 15 |- ( ( (FermatNo ` n ) = F /\ (FermatNo ` m ) = G ) -> ( -. (FermatNo ` n ) = (FermatNo ` m ) <-> -. F = G ) )
28		oveq12 6659	. . . . . . . . . . . . . . . 16 |- ( ( (FermatNo ` n ) = F /\ (FermatNo ` m ) = G ) -> ( (FermatNo ` n ) gcd (FermatNo ` m ) ) = ( F gcd G ) )
29	28	eqeq1d 2624	. . . . . . . . . . . . . . 15 |- ( ( (FermatNo ` n ) = F /\ (FermatNo ` m ) = G ) -> ( ( (FermatNo ` n ) gcd (FermatNo ` m ) ) = 1 <-> ( F gcd G ) = 1 ) )
30	27, 29	imbi12d 334	. . . . . . . . . . . . . 14 |- ( ( (FermatNo ` n ) = F /\ (FermatNo ` m ) = G ) -> ( ( -. (FermatNo ` n ) = (FermatNo ` m ) -> ( (FermatNo ` n ) gcd (FermatNo ` m ) ) = 1 ) <-> ( -. F = G -> ( F gcd G ) = 1 ) ) )
31	30	ancoms 469	. . . . . . . . . . . . 13 |- ( ( (FermatNo ` m ) = G /\ (FermatNo ` n ) = F ) -> ( ( -. (FermatNo ` n ) = (FermatNo ` m ) -> ( (FermatNo ` n ) gcd (FermatNo ` m ) ) = 1 ) <-> ( -. F = G -> ( F gcd G ) = 1 ) ) )
32	25, 31	syl5ibcom 235	. . . . . . . . . . . 12 |- ( ( n e. NN0 /\ m e. NN0 ) -> ( ( (FermatNo ` m ) = G /\ (FermatNo ` n ) = F ) -> ( -. F = G -> ( F gcd G ) = 1 ) ) )
33	32	com23 86	. . . . . . . . . . 11 |- ( ( n e. NN0 /\ m e. NN0 ) -> ( -. F = G -> ( ( (FermatNo ` m ) = G /\ (FermatNo ` n ) = F ) -> ( F gcd G ) = 1 ) ) )
34	33	impcom 446	. . . . . . . . . 10 |- ( ( -. F = G /\ ( n e. NN0 /\ m e. NN0 ) ) -> ( ( (FermatNo ` m ) = G /\ (FermatNo ` n ) = F ) -> ( F gcd G ) = 1 ) )
35	34	imp 445	. . . . . . . . 9 |- ( ( ( -. F = G /\ ( n e. NN0 /\ m e. NN0 ) ) /\ ( (FermatNo ` m ) = G /\ (FermatNo ` n ) = F ) ) -> ( F gcd G ) = 1 )
36		prmnn 15388	. . . . . . . . . . . 12 |- ( I e. Prime -> I e. NN )
37		coprmdvds1 15365	. . . . . . . . . . . . 13 |- ( ( F e. NN /\ G e. NN /\ ( F gcd G ) = 1 ) -> ( ( I e. NN /\ I || F /\ I || G ) -> I = 1 ) )
38	37	imp 445	. . . . . . . . . . . 12 |- ( ( ( F e. NN /\ G e. NN /\ ( F gcd G ) = 1 ) /\ ( I e. NN /\ I || F /\ I || G ) ) -> I = 1 )
39	36, 38	syl3anr1 1378	. . . . . . . . . . 11 |- ( ( ( F e. NN /\ G e. NN /\ ( F gcd G ) = 1 ) /\ ( I e. Prime /\ I || F /\ I || G ) ) -> I = 1 )
40		eleq1 2689	. . . . . . . . . . . . . . . 16 |- ( I = 1 -> ( I e. Prime <-> 1 e. Prime ) )
41		1nprm 15392	. . . . . . . . . . . . . . . . 17 |- -. 1 e. Prime
42	41	pm2.21i 116	. . . . . . . . . . . . . . . 16 |- ( 1 e. Prime -> F = G )
43	40, 42	syl6bi 243	. . . . . . . . . . . . . . 15 |- ( I = 1 -> ( I e. Prime -> F = G ) )
44	43	com12 32	. . . . . . . . . . . . . 14 |- ( I e. Prime -> ( I = 1 -> F = G ) )
45	44	a1d 25	. . . . . . . . . . . . 13 |- ( I e. Prime -> ( ( F e. NN /\ G e. NN /\ ( F gcd G ) = 1 ) -> ( I = 1 -> F = G ) ) )
46	45	3ad2ant1 1082	. . . . . . . . . . . 12 |- ( ( I e. Prime /\ I || F /\ I || G ) -> ( ( F e. NN /\ G e. NN /\ ( F gcd G ) = 1 ) -> ( I = 1 -> F = G ) ) )
47	46	impcom 446	. . . . . . . . . . 11 |- ( ( ( F e. NN /\ G e. NN /\ ( F gcd G ) = 1 ) /\ ( I e. Prime /\ I || F /\ I || G ) ) -> ( I = 1 -> F = G ) )
48	39, 47	mpd 15	. . . . . . . . . 10 |- ( ( ( F e. NN /\ G e. NN /\ ( F gcd G ) = 1 ) /\ ( I e. Prime /\ I || F /\ I || G ) ) -> F = G )
49	48	ex 450	. . . . . . . . 9 |- ( ( F e. NN /\ G e. NN /\ ( F gcd G ) = 1 ) -> ( ( I e. Prime /\ I || F /\ I || G ) -> F = G ) )
50	10, 16, 35, 49	syl3anc 1326	. . . . . . . 8 |- ( ( ( -. F = G /\ ( n e. NN0 /\ m e. NN0 ) ) /\ ( (FermatNo ` m ) = G /\ (FermatNo ` n ) = F ) ) -> ( ( I e. Prime /\ I || F /\ I || G ) -> F = G ) )
51	50	exp43 640	. . . . . . 7 |- ( -. F = G -> ( ( n e. NN0 /\ m e. NN0 ) -> ( (FermatNo ` m ) = G -> ( (FermatNo ` n ) = F -> ( ( I e. Prime /\ I || F /\ I || G ) -> F = G ) ) ) ) )
52	4, 51	pm2.61i 176	. . . . . 6 |- ( ( n e. NN0 /\ m e. NN0 ) -> ( (FermatNo ` m ) = G -> ( (FermatNo ` n ) = F -> ( ( I e. Prime /\ I || F /\ I || G ) -> F = G ) ) ) )
53	52	rexlimdva 3031	. . . . 5 |- ( n e. NN0 -> ( E. m e. NN0 (FermatNo ` m ) = G -> ( (FermatNo ` n ) = F -> ( ( I e. Prime /\ I || F /\ I || G ) -> F = G ) ) ) )
54	53	com23 86	. . . 4 |- ( n e. NN0 -> ( (FermatNo ` n ) = F -> ( E. m e. NN0 (FermatNo ` m ) = G -> ( ( I e. Prime /\ I || F /\ I || G ) -> F = G ) ) ) )
55	54	rexlimiv 3027	. . 3 |- ( E. n e. NN0 (FermatNo ` n ) = F -> ( E. m e. NN0 (FermatNo ` m ) = G -> ( ( I e. Prime /\ I || F /\ I || G ) -> F = G ) ) )
56	55	imp 445	. 2 |- ( ( E. n e. NN0 (FermatNo ` n ) = F /\ E. m e. NN0 (FermatNo ` m ) = G ) -> ( ( I e. Prime /\ I || F /\ I || G ) -> F = G ) )
57	1, 2, 56	syl2anb 496	1 |- ( ( F e. ran FermatNo /\ G e. ran FermatNo ) -> ( ( I e. Prime /\ I || F /\ I || G ) -> F = G ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ran crn 5115   ` cfv 5888  (class class class)co 6650   1c1 9937   NNcn 11020   NN0cn0 11292   || cdvds 14983   gcd cgcd 15216   Primecprime 15385  FermatNocfmtno 41439

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-inf2 8538  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-fal 1489  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-prod 14636  df-dvds 14984  df-gcd 15217  df-prm 15386  df-fmtno 41440

This theorem is referenced by:  prmdvdsfmtnof1  41499