Theorem prmdvdsfmtnof1lem1 41496

Description: Lemma 1 for prmdvdsfmtnof1 41499. (Contributed by AV, 3-Aug-2021.)

Hypotheses

Ref	Expression
prmdvdsfmtnof1lem1.i	|- I = inf ( { p e. Prime | p || F } , RR , < )
prmdvdsfmtnof1lem1.j	|- J = inf ( { p e. Prime | p || G } , RR , < )

Assertion

Ref	Expression
prmdvdsfmtnof1lem1	|- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> ( I = J -> ( I e. Prime /\ I || F /\ I || G ) ) )

Distinct variable groups:   F, p   G, p

Allowed substitution hints:   I( p)   J( p)

Proof of Theorem prmdvdsfmtnof1lem1

Step	Hyp	Ref	Expression
1		ltso 10118	. . . 4 |- < Or RR
2	1	a1i 11	. . 3 |- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> < Or RR )
3		eluz2nn 11726	. . . . 5 |- ( F e. ( ZZ>= ` 2 ) -> F e. NN )
4	3	adantr 481	. . . 4 |- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> F e. NN )
5		prmdvdsfi 24833	. . . 4 |- ( F e. NN -> { p e. Prime | p || F } e. Fin )
6	4, 5	syl 17	. . 3 |- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> { p e. Prime | p || F } e. Fin )
7		exprmfct 15416	. . . . 5 |- ( F e. ( ZZ>= ` 2 ) -> E. p e. Prime p || F )
8	7	adantr 481	. . . 4 |- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> E. p e. Prime p || F )
9		rabn0 3958	. . . 4 |- ( { p e. Prime | p || F } =/= (/) <-> E. p e. Prime p || F )
10	8, 9	sylibr 224	. . 3 |- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> { p e. Prime | p || F } =/= (/) )
11		ssrab2 3687	. . . . 5 |- { p e. Prime | p || F } C_ Prime
12		prmssnn 15390	. . . . . 6 |- Prime C_ NN
13		nnssre 11024	. . . . . 6 |- NN C_ RR
14	12, 13	sstri 3612	. . . . 5 |- Prime C_ RR
15	11, 14	sstri 3612	. . . 4 |- { p e. Prime | p || F } C_ RR
16	15	a1i 11	. . 3 |- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> { p e. Prime | p || F } C_ RR )
17		fiinfcl 8407	. . 3 |- ( ( < Or RR /\ ( { p e. Prime | p || F } e. Fin /\ { p e. Prime | p || F } =/= (/) /\ { p e. Prime | p || F } C_ RR ) ) -> inf ( { p e. Prime | p || F } , RR , < ) e. { p e. Prime | p || F } )
18	2, 6, 10, 16, 17	syl13anc 1328	. 2 |- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> inf ( { p e. Prime | p || F } , RR , < ) e. { p e. Prime | p || F } )
19		prmdvdsfmtnof1lem1.i	. . . 4 |- I = inf ( { p e. Prime | p || F } , RR , < )
20	19	eleq1i 2692	. . 3 |- ( I e. { p e. Prime | p || F } <-> inf ( { p e. Prime | p || F } , RR , < ) e. { p e. Prime | p || F } )
21		eluz2nn 11726	. . . . . . 7 |- ( G e. ( ZZ>= ` 2 ) -> G e. NN )
22	21	adantl 482	. . . . . 6 |- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> G e. NN )
23		prmdvdsfi 24833	. . . . . 6 |- ( G e. NN -> { p e. Prime | p || G } e. Fin )
24	22, 23	syl 17	. . . . 5 |- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> { p e. Prime | p || G } e. Fin )
25		exprmfct 15416	. . . . . . 7 |- ( G e. ( ZZ>= ` 2 ) -> E. p e. Prime p || G )
26	25	adantl 482	. . . . . 6 |- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> E. p e. Prime p || G )
27		rabn0 3958	. . . . . 6 |- ( { p e. Prime | p || G } =/= (/) <-> E. p e. Prime p || G )
28	26, 27	sylibr 224	. . . . 5 |- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> { p e. Prime | p || G } =/= (/) )
29		ssrab2 3687	. . . . . . 7 |- { p e. Prime | p || G } C_ Prime
30	29, 14	sstri 3612	. . . . . 6 |- { p e. Prime | p || G } C_ RR
31	30	a1i 11	. . . . 5 |- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> { p e. Prime | p || G } C_ RR )
32		fiinfcl 8407	. . . . 5 |- ( ( < Or RR /\ ( { p e. Prime | p || G } e. Fin /\ { p e. Prime | p || G } =/= (/) /\ { p e. Prime | p || G } C_ RR ) ) -> inf ( { p e. Prime | p || G } , RR , < ) e. { p e. Prime | p || G } )
33	2, 24, 28, 31, 32	syl13anc 1328	. . . 4 |- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> inf ( { p e. Prime | p || G } , RR , < ) e. { p e. Prime | p || G } )
34		prmdvdsfmtnof1lem1.j	. . . . . 6 |- J = inf ( { p e. Prime | p || G } , RR , < )
35	34	eleq1i 2692	. . . . 5 |- ( J e. { p e. Prime | p || G } <-> inf ( { p e. Prime | p || G } , RR , < ) e. { p e. Prime | p || G } )
36		nfrab1 3122	. . . . . . . . . 10 |- F/_ p { p e. Prime | p || G }
37		nfcv 2764	. . . . . . . . . 10 |- F/_ p RR
38		nfcv 2764	. . . . . . . . . 10 |- F/_ p <
39	36, 37, 38	nfinf 8388	. . . . . . . . 9 |- F/_ pinf ( { p e. Prime | p || G } , RR , < )
40	34, 39	nfcxfr 2762	. . . . . . . 8 |- F/_ p J
41		nfcv 2764	. . . . . . . 8 |- F/_ p Prime
42		nfcv 2764	. . . . . . . . 9 |- F/_ p ||
43		nfcv 2764	. . . . . . . . 9 |- F/_ p G
44	40, 42, 43	nfbr 4699	. . . . . . . 8 |- F/ p J || G
45		breq1 4656	. . . . . . . 8 |- ( p = J -> ( p || G <-> J || G ) )
46	40, 41, 44, 45	elrabf 3360	. . . . . . 7 |- ( J e. { p e. Prime | p || G } <-> ( J e. Prime /\ J || G ) )
47		nfrab1 3122	. . . . . . . . . . 11 |- F/_ p { p e. Prime | p || F }
48	47, 37, 38	nfinf 8388	. . . . . . . . . 10 |- F/_ pinf ( { p e. Prime | p || F } , RR , < )
49	19, 48	nfcxfr 2762	. . . . . . . . 9 |- F/_ p I
50		nfcv 2764	. . . . . . . . . 10 |- F/_ p F
51	49, 42, 50	nfbr 4699	. . . . . . . . 9 |- F/ p I || F
52		breq1 4656	. . . . . . . . 9 |- ( p = I -> ( p || F <-> I || F ) )
53	49, 41, 51, 52	elrabf 3360	. . . . . . . 8 |- ( I e. { p e. Prime | p || F } <-> ( I e. Prime /\ I || F ) )
54		simp2l 1087	. . . . . . . . . 10 |- ( ( ( J e. Prime /\ J || G ) /\ ( I e. Prime /\ I || F ) /\ I = J ) -> I e. Prime )
55		simp2r 1088	. . . . . . . . . 10 |- ( ( ( J e. Prime /\ J || G ) /\ ( I e. Prime /\ I || F ) /\ I = J ) -> I || F )
56		simp1r 1086	. . . . . . . . . . 11 |- ( ( ( J e. Prime /\ J || G ) /\ ( I e. Prime /\ I || F ) /\ I = J ) -> J || G )
57		breq1 4656	. . . . . . . . . . . 12 |- ( I = J -> ( I || G <-> J || G ) )
58	57	3ad2ant3 1084	. . . . . . . . . . 11 |- ( ( ( J e. Prime /\ J || G ) /\ ( I e. Prime /\ I || F ) /\ I = J ) -> ( I || G <-> J || G ) )
59	56, 58	mpbird 247	. . . . . . . . . 10 |- ( ( ( J e. Prime /\ J || G ) /\ ( I e. Prime /\ I || F ) /\ I = J ) -> I || G )
60	54, 55, 59	3jca 1242	. . . . . . . . 9 |- ( ( ( J e. Prime /\ J || G ) /\ ( I e. Prime /\ I || F ) /\ I = J ) -> ( I e. Prime /\ I || F /\ I || G ) )
61	60	3exp 1264	. . . . . . . 8 |- ( ( J e. Prime /\ J || G ) -> ( ( I e. Prime /\ I || F ) -> ( I = J -> ( I e. Prime /\ I || F /\ I || G ) ) ) )
62	53, 61	syl5bi 232	. . . . . . 7 |- ( ( J e. Prime /\ J || G ) -> ( I e. { p e. Prime | p || F } -> ( I = J -> ( I e. Prime /\ I || F /\ I || G ) ) ) )
63	46, 62	sylbi 207	. . . . . 6 |- ( J e. { p e. Prime | p || G } -> ( I e. { p e. Prime | p || F } -> ( I = J -> ( I e. Prime /\ I || F /\ I || G ) ) ) )
64	63	a1i 11	. . . . 5 |- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> ( J e. { p e. Prime | p || G } -> ( I e. { p e. Prime | p || F } -> ( I = J -> ( I e. Prime /\ I || F /\ I || G ) ) ) ) )
65	35, 64	syl5bir 233	. . . 4 |- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> (inf ( { p e. Prime | p || G } , RR , < ) e. { p e. Prime | p || G } -> ( I e. { p e. Prime | p || F } -> ( I = J -> ( I e. Prime /\ I || F /\ I || G ) ) ) ) )
66	33, 65	mpd 15	. . 3 |- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> ( I e. { p e. Prime | p || F } -> ( I = J -> ( I e. Prime /\ I || F /\ I || G ) ) ) )
67	20, 66	syl5bir 233	. 2 |- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> (inf ( { p e. Prime | p || F } , RR , < ) e. { p e. Prime | p || F } -> ( I = J -> ( I e. Prime /\ I || F /\ I || G ) ) ) )
68	18, 67	mpd 15	1 |- ( ( F e. ( ZZ>= ` 2 ) /\ G e. ( ZZ>= ` 2 ) ) -> ( I = J -> ( I e. Prime /\ I || F /\ I || G ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   C_ wss 3574   (/)c0 3915   class class class wbr 4653   Or wor 5034   ` cfv 5888   Fincfn 7955  infcinf 8347   RRcr 9935   < clt 10074   NNcn 11020   2c2 11070   ZZ>=cuz 11687   || cdvds 14983   Primecprime 15385

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-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

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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  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

This theorem is referenced by:  prmdvdsfmtnof1  41499