Theorem goldbachth 41459

Description: Goldbach's theorem: Two different Fermat numbers are coprime. See ProofWiki "Goldbach's theorem", 31-Jul-2021, https://proofwiki.org/wiki/Goldbach%27s_Theorem or Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 1-Aug-2021.)

Assertion

Ref	Expression
goldbachth	|- ( ( N e. NN0 /\ M e. NN0 /\ N =/= M ) -> ( (FermatNo ` N ) gcd (FermatNo ` M ) ) = 1 )

Proof of Theorem goldbachth

Step	Hyp	Ref	Expression
1		nn0re 11301	. . . 4 |- ( N e. NN0 -> N e. RR )
2		nn0re 11301	. . . 4 |- ( M e. NN0 -> M e. RR )
3		lttri4 10122	. . . 4 |- ( ( N e. RR /\ M e. RR ) -> ( N < M \/ N = M \/ M < N ) )
4	1, 2, 3	syl2an 494	. . 3 |- ( ( N e. NN0 /\ M e. NN0 ) -> ( N < M \/ N = M \/ M < N ) )
5	4	3adant3 1081	. 2 |- ( ( N e. NN0 /\ M e. NN0 /\ N =/= M ) -> ( N < M \/ N = M \/ M < N ) )
6		fmtnonn 41443	. . . . . . . . . 10 |- ( N e. NN0 -> (FermatNo ` N ) e. NN )
7	6	nnzd 11481	. . . . . . . . 9 |- ( N e. NN0 -> (FermatNo ` N ) e. ZZ )
8		fmtnonn 41443	. . . . . . . . . 10 |- ( M e. NN0 -> (FermatNo ` M ) e. NN )
9	8	nnzd 11481	. . . . . . . . 9 |- ( M e. NN0 -> (FermatNo ` M ) e. ZZ )
10		gcdcom 15235	. . . . . . . . 9 |- ( ( (FermatNo ` N ) e. ZZ /\ (FermatNo ` M ) e. ZZ ) -> ( (FermatNo ` N ) gcd (FermatNo ` M ) ) = ( (FermatNo ` M ) gcd (FermatNo ` N ) ) )
11	7, 9, 10	syl2anr 495	. . . . . . . 8 |- ( ( M e. NN0 /\ N e. NN0 ) -> ( (FermatNo ` N ) gcd (FermatNo ` M ) ) = ( (FermatNo ` M ) gcd (FermatNo ` N ) ) )
12	11	3adant3 1081	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 /\ N < M ) -> ( (FermatNo ` N ) gcd (FermatNo ` M ) ) = ( (FermatNo ` M ) gcd (FermatNo ` N ) ) )
13		goldbachthlem2 41458	. . . . . . 7 |- ( ( M e. NN0 /\ N e. NN0 /\ N < M ) -> ( (FermatNo ` M ) gcd (FermatNo ` N ) ) = 1 )
14	12, 13	eqtrd 2656	. . . . . 6 |- ( ( M e. NN0 /\ N e. NN0 /\ N < M ) -> ( (FermatNo ` N ) gcd (FermatNo ` M ) ) = 1 )
15	14	3exp 1264	. . . . 5 |- ( M e. NN0 -> ( N e. NN0 -> ( N < M -> ( (FermatNo ` N ) gcd (FermatNo ` M ) ) = 1 ) ) )
16	15	impcom 446	. . . 4 |- ( ( N e. NN0 /\ M e. NN0 ) -> ( N < M -> ( (FermatNo ` N ) gcd (FermatNo ` M ) ) = 1 ) )
17	16	3adant3 1081	. . 3 |- ( ( N e. NN0 /\ M e. NN0 /\ N =/= M ) -> ( N < M -> ( (FermatNo ` N ) gcd (FermatNo ` M ) ) = 1 ) )
18		eqneqall 2805	. . . . 5 |- ( N = M -> ( N =/= M -> ( (FermatNo ` N ) gcd (FermatNo ` M ) ) = 1 ) )
19	18	com12 32	. . . 4 |- ( N =/= M -> ( N = M -> ( (FermatNo ` N ) gcd (FermatNo ` M ) ) = 1 ) )
20	19	3ad2ant3 1084	. . 3 |- ( ( N e. NN0 /\ M e. NN0 /\ N =/= M ) -> ( N = M -> ( (FermatNo ` N ) gcd (FermatNo ` M ) ) = 1 ) )
21		goldbachthlem2 41458	. . . . 5 |- ( ( N e. NN0 /\ M e. NN0 /\ M < N ) -> ( (FermatNo ` N ) gcd (FermatNo ` M ) ) = 1 )
22	21	3expia 1267	. . . 4 |- ( ( N e. NN0 /\ M e. NN0 ) -> ( M < N -> ( (FermatNo ` N ) gcd (FermatNo ` M ) ) = 1 ) )
23	22	3adant3 1081	. . 3 |- ( ( N e. NN0 /\ M e. NN0 /\ N =/= M ) -> ( M < N -> ( (FermatNo ` N ) gcd (FermatNo ` M ) ) = 1 ) )
24	17, 20, 23	3jaod 1392	. 2 |- ( ( N e. NN0 /\ M e. NN0 /\ N =/= M ) -> ( ( N < M \/ N = M \/ M < N ) -> ( (FermatNo ` N ) gcd (FermatNo ` M ) ) = 1 ) )
25	5, 24	mpd 15	1 |- ( ( N e. NN0 /\ M e. NN0 /\ N =/= M ) -> ( (FermatNo ` N ) gcd (FermatNo ` M ) ) = 1 )

Syntax hints:   -> wi 4   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   < clt 10074   NN0cn0 11292   ZZcz 11377   gcd cgcd 15216  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:  prmdvdsfmtnof1lem2  41497