Theorem exprmfct 12331

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 9657	. 2 |- ( N e. ( ZZ>= ` 2 ) -> N e. NN )
2		eleq1 2259	. . . 4 |- ( x = 1 -> ( x e. ( ZZ>= ` 2 ) <-> 1 e. ( ZZ>= ` 2 ) ) )
3	2	imbi1d 231	. . 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 2259	. . . 4 |- ( x = y -> ( x e. ( ZZ>= ` 2 ) <-> y e. ( ZZ>= ` 2 ) ) )
5		breq2 4038	. . . . 5 |- ( x = y -> ( p || x <-> p || y ) )
6	5	rexbidv 2498	. . . 4 |- ( x = y -> ( E. p e. Prime p || x <-> E. p e. Prime p || y ) )
7	4, 6	imbi12d 234	. . 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 2259	. . . 4 |- ( x = z -> ( x e. ( ZZ>= ` 2 ) <-> z e. ( ZZ>= ` 2 ) ) )
9		breq2 4038	. . . . 5 |- ( x = z -> ( p || x <-> p || z ) )
10	9	rexbidv 2498	. . . 4 |- ( x = z -> ( E. p e. Prime p || x <-> E. p e. Prime p || z ) )
11	8, 10	imbi12d 234	. . 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 2259	. . . 4 |- ( x = ( y x. z ) -> ( x e. ( ZZ>= ` 2 ) <-> ( y x. z ) e. ( ZZ>= ` 2 ) ) )
13		breq2 4038	. . . . 5 |- ( x = ( y x. z ) -> ( p || x <-> p || ( y x. z ) ) )
14	13	rexbidv 2498	. . . 4 |- ( x = ( y x. z ) -> ( E. p e. Prime p || x <-> E. p e. Prime p || ( y x. z ) ) )
15	12, 14	imbi12d 234	. . 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 2259	. . . 4 |- ( x = N -> ( x e. ( ZZ>= ` 2 ) <-> N e. ( ZZ>= ` 2 ) ) )
17		breq2 4038	. . . . 5 |- ( x = N -> ( p || x <-> p || N ) )
18	17	rexbidv 2498	. . . 4 |- ( x = N -> ( E. p e. Prime p || x <-> E. p e. Prime p || N ) )
19	16, 18	imbi12d 234	. . 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 9076	. . . . 5 |- ( 1 - 1 ) = 0
21		uz2m1nn 9696	. . . . 5 |- ( 1 e. ( ZZ>= ` 2 ) -> ( 1 - 1 ) e. NN )
22	20, 21	eqeltrrid 2284	. . . 4 |- ( 1 e. ( ZZ>= ` 2 ) -> 0 e. NN )
23		0nnn 9034	. . . . 5 |- -. 0 e. NN
24	23	pm2.21i 647	. . . 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 12304	. . . . . 6 |- ( x e. Prime -> x e. ZZ )
27		iddvds 11986	. . . . . 6 |- ( x e. ZZ -> x || x )
28	26, 27	syl 14	. . . . 5 |- ( x e. Prime -> x || x )
29		breq1 4037	. . . . . 6 |- ( p = x -> ( p || x <-> x || x ) )
30	29	rspcev 2868	. . . . 5 |- ( ( x e. Prime /\ x || x ) -> E. p e. Prime p || x )
31	28, 30	mpdan 421	. . . 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 109	. . . . . 6 |- ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) -> y e. ( ZZ>= ` 2 ) )
34		eluzelz 9627	. . . . . . . . . 10 |- ( y e. ( ZZ>= ` 2 ) -> y e. ZZ )
35	34	ad2antrr 488	. . . . . . . . 9 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> y e. ZZ )
36		eluzelz 9627	. . . . . . . . . 10 |- ( z e. ( ZZ>= ` 2 ) -> z e. ZZ )
37	36	ad2antlr 489	. . . . . . . . 9 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> z e. ZZ )
38		dvdsmul1 11995	. . . . . . . . 9 |- ( ( y e. ZZ /\ z e. ZZ ) -> y || ( y x. z ) )
39	35, 37, 38	syl2anc 411	. . . . . . . 8 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> y || ( y x. z ) )
40		prmz 12304	. . . . . . . . . 10 |- ( p e. Prime -> p e. ZZ )
41	40	adantl 277	. . . . . . . . 9 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> p e. ZZ )
42	35, 37	zmulcld 9471	. . . . . . . . 9 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> ( y x. z ) e. ZZ )
43		dvdstr 12010	. . . . . . . . 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 1249	. . . . . . . 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 428	. . . . . . 7 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> ( p || y -> p || ( y x. z ) ) )
46	45	reximdva 2599	. . . . . 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 279	. . 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 12314	. 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 104   = wceq 1364   e. wcel 2167   E.wrex 2476   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   0cc0 7896   1c1 7897   x. cmul 7901   - cmin 8214   NNcn 9007   2c2 9058   ZZcz 9343   ZZ>=cuz 9618   || cdvds 11969   Primecprime 12300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-2o 6484  df-er 6601  df-en 6809  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-fl 10377  df-mod 10432  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-dvds 11970  df-prm 12301

This theorem is referenced by:  prmdvdsfz  12332  isprm5lem  12334  rpexp  12346  pc2dvds  12524  oddprmdvds  12548  prmunb  12556  lgsne0  15363