Theorem exprmfct 12092

Description: Every integer greater than or equal to 2 has a prime factor. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 20-Jun-2015.)

Assertion

Ref	Expression
exprmfct	|- ( N e. ( ZZ>= ` 2 ) -> E. p e. Prime p || N )

Distinct variable group:   N, p

Proof of Theorem exprmfct

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

Step	Hyp	Ref	Expression
1		eluz2nn 9525	. 2 |- ( N e. ( ZZ>= ` 2 ) -> N e. NN )
2		eleq1 2233	. . . 4 |- ( x = 1 -> ( x e. ( ZZ>= ` 2 ) <-> 1 e. ( ZZ>= ` 2 ) ) )
3	2	imbi1d 230	. . 3 |- ( x = 1 -> ( ( x e. ( ZZ>= ` 2 ) -> E. p e. Prime p || x ) <-> ( 1 e. ( ZZ>= ` 2 ) -> E. p e. Prime p || x ) ) )
4		eleq1 2233	. . . 4 |- ( x = y -> ( x e. ( ZZ>= ` 2 ) <-> y e. ( ZZ>= ` 2 ) ) )
5		breq2 3993	. . . . 5 |- ( x = y -> ( p || x <-> p || y ) )
6	5	rexbidv 2471	. . . 4 |- ( x = y -> ( E. p e. Prime p || x <-> E. p e. Prime p || y ) )
7	4, 6	imbi12d 233	. . 3 |- ( x = y -> ( ( x e. ( ZZ>= ` 2 ) -> E. p e. Prime p || x ) <-> ( y e. ( ZZ>= ` 2 ) -> E. p e. Prime p || y ) ) )
8		eleq1 2233	. . . 4 |- ( x = z -> ( x e. ( ZZ>= ` 2 ) <-> z e. ( ZZ>= ` 2 ) ) )
9		breq2 3993	. . . . 5 |- ( x = z -> ( p || x <-> p || z ) )
10	9	rexbidv 2471	. . . 4 |- ( x = z -> ( E. p e. Prime p || x <-> E. p e. Prime p || z ) )
11	8, 10	imbi12d 233	. . 3 |- ( x = z -> ( ( x e. ( ZZ>= ` 2 ) -> E. p e. Prime p || x ) <-> ( z e. ( ZZ>= ` 2 ) -> E. p e. Prime p || z ) ) )
12		eleq1 2233	. . . 4 |- ( x = ( y x. z ) -> ( x e. ( ZZ>= ` 2 ) <-> ( y x. z ) e. ( ZZ>= ` 2 ) ) )
13		breq2 3993	. . . . 5 |- ( x = ( y x. z ) -> ( p || x <-> p || ( y x. z ) ) )
14	13	rexbidv 2471	. . . 4 |- ( x = ( y x. z ) -> ( E. p e. Prime p || x <-> E. p e. Prime p || ( y x. z ) ) )
15	12, 14	imbi12d 233	. . 3 |- ( x = ( y x. z ) -> ( ( x e. ( ZZ>= ` 2 ) -> E. p e. Prime p || x ) <-> ( ( y x. z ) e. ( ZZ>= ` 2 ) -> E. p e. Prime p || ( y x. z ) ) ) )
16		eleq1 2233	. . . 4 |- ( x = N -> ( x e. ( ZZ>= ` 2 ) <-> N e. ( ZZ>= ` 2 ) ) )
17		breq2 3993	. . . . 5 |- ( x = N -> ( p || x <-> p || N ) )
18	17	rexbidv 2471	. . . 4 |- ( x = N -> ( E. p e. Prime p || x <-> E. p e. Prime p || N ) )
19	16, 18	imbi12d 233	. . 3 |- ( x = N -> ( ( x e. ( ZZ>= ` 2 ) -> E. p e. Prime p || x ) <-> ( N e. ( ZZ>= ` 2 ) -> E. p e. Prime p || N ) ) )
20		1m1e0 8947	. . . . 5 |- ( 1 - 1 ) = 0
21		uz2m1nn 9564	. . . . 5 |- ( 1 e. ( ZZ>= ` 2 ) -> ( 1 - 1 ) e. NN )
22	20, 21	eqeltrrid 2258	. . . 4 |- ( 1 e. ( ZZ>= ` 2 ) -> 0 e. NN )
23		0nnn 8905	. . . . 5 |- -. 0 e. NN
24	23	pm2.21i 641	. . . 4 |- ( 0 e. NN -> E. p e. Prime p || x )
25	22, 24	syl 14	. . 3 |- ( 1 e. ( ZZ>= ` 2 ) -> E. p e. Prime p || x )
26		prmz 12065	. . . . . 6 |- ( x e. Prime -> x e. ZZ )
27		iddvds 11766	. . . . . 6 |- ( x e. ZZ -> x || x )
28	26, 27	syl 14	. . . . 5 |- ( x e. Prime -> x || x )
29		breq1 3992	. . . . . 6 |- ( p = x -> ( p || x <-> x || x ) )
30	29	rspcev 2834	. . . . 5 |- ( ( x e. Prime /\ x || x ) -> E. p e. Prime p || x )
31	28, 30	mpdan 419	. . . 4 |- ( x e. Prime -> E. p e. Prime p || x )
32	31	a1d 22	. . 3 |- ( x e. Prime -> ( x e. ( ZZ>= ` 2 ) -> E. p e. Prime p || x ) )
33		simpl 108	. . . . . 6 |- ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) -> y e. ( ZZ>= ` 2 ) )
34		eluzelz 9496	. . . . . . . . . 10 |- ( y e. ( ZZ>= ` 2 ) -> y e. ZZ )
35	34	ad2antrr 485	. . . . . . . . 9 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> y e. ZZ )
36		eluzelz 9496	. . . . . . . . . 10 |- ( z e. ( ZZ>= ` 2 ) -> z e. ZZ )
37	36	ad2antlr 486	. . . . . . . . 9 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> z e. ZZ )
38		dvdsmul1 11775	. . . . . . . . 9 |- ( ( y e. ZZ /\ z e. ZZ ) -> y || ( y x. z ) )
39	35, 37, 38	syl2anc 409	. . . . . . . 8 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> y || ( y x. z ) )
40		prmz 12065	. . . . . . . . . 10 |- ( p e. Prime -> p e. ZZ )
41	40	adantl 275	. . . . . . . . 9 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> p e. ZZ )
42	35, 37	zmulcld 9340	. . . . . . . . 9 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> ( y x. z ) e. ZZ )
43		dvdstr 11790	. . . . . . . . 9 |- ( ( p e. ZZ /\ y e. ZZ /\ ( y x. z ) e. ZZ ) -> ( ( p || y /\ y || ( y x. z ) ) -> p || ( y x. z ) ) )
44	41, 35, 42, 43	syl3anc 1233	. . . . . . . 8 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> ( ( p || y /\ y || ( y x. z ) ) -> p || ( y x. z ) ) )
45	39, 44	mpan2d 426	. . . . . . 7 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> ( p || y -> p || ( y x. z ) ) )
46	45	reximdva 2572	. . . . . 6 |- ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) -> ( E. p e. Prime p || y -> E. p e. Prime p || ( y x. z ) ) )
47	33, 46	embantd 56	. . . . 5 |- ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) -> ( ( y e. ( ZZ>= ` 2 ) -> E. p e. Prime p || y ) -> E. p e. Prime p || ( y x. z ) ) )
48	47	a1dd 48	. . . 4 |- ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) -> ( ( y e. ( ZZ>= ` 2 ) -> E. p e. Prime p || y ) -> ( ( y x. z ) e. ( ZZ>= ` 2 ) -> E. p e. Prime p || ( y x. z ) ) ) )
49	48	adantrd 277	. . 3 |- ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) -> ( ( ( y e. ( ZZ>= ` 2 ) -> E. p e. Prime p || y ) /\ ( z e. ( ZZ>= ` 2 ) -> E. p e. Prime p || z ) ) -> ( ( y x. z ) e. ( ZZ>= ` 2 ) -> E. p e. Prime p || ( y x. z ) ) ) )
50	3, 7, 11, 15, 19, 25, 32, 49	prmind 12075	. 2 |- ( N e. NN -> ( N e. ( ZZ>= ` 2 ) -> E. p e. Prime p || N ) )
51	1, 50	mpcom 36	1 |- ( N e. ( ZZ>= ` 2 ) -> E. p e. Prime p || N )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   E.wrex 2449   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   0cc0 7774   1c1 7775   x. cmul 7779   - cmin 8090   NNcn 8878   2c2 8929   ZZcz 9212   ZZ>=cuz 9487   || cdvds 11749   Primecprime 12061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-1o 6395  df-2o 6396  df-er 6513  df-en 6719  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-prm 12062

This theorem is referenced by:  prmdvdsfz  12093  isprm5lem  12095  rpexp  12107  pc2dvds  12283  oddprmdvds  12306  prmunb  12314  lgsne0  13733