Theorem exprmfct 12070

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 9504	. 2 |- ( N e. ( ZZ>= ` 2 ) -> N e. NN )
2		eleq1 2229	. . . 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 2229	. . . 4 |- ( x = y -> ( x e. ( ZZ>= ` 2 ) <-> y e. ( ZZ>= ` 2 ) ) )
5		breq2 3986	. . . . 5 |- ( x = y -> ( p || x <-> p || y ) )
6	5	rexbidv 2467	. . . 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 2229	. . . 4 |- ( x = z -> ( x e. ( ZZ>= ` 2 ) <-> z e. ( ZZ>= ` 2 ) ) )
9		breq2 3986	. . . . 5 |- ( x = z -> ( p || x <-> p || z ) )
10	9	rexbidv 2467	. . . 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 2229	. . . 4 |- ( x = ( y x. z ) -> ( x e. ( ZZ>= ` 2 ) <-> ( y x. z ) e. ( ZZ>= ` 2 ) ) )
13		breq2 3986	. . . . 5 |- ( x = ( y x. z ) -> ( p || x <-> p || ( y x. z ) ) )
14	13	rexbidv 2467	. . . 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 2229	. . . 4 |- ( x = N -> ( x e. ( ZZ>= ` 2 ) <-> N e. ( ZZ>= ` 2 ) ) )
17		breq2 3986	. . . . 5 |- ( x = N -> ( p || x <-> p || N ) )
18	17	rexbidv 2467	. . . 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 8926	. . . . 5 |- ( 1 - 1 ) = 0
21		uz2m1nn 9543	. . . . 5 |- ( 1 e. ( ZZ>= ` 2 ) -> ( 1 - 1 ) e. NN )
22	20, 21	eqeltrrid 2254	. . . 4 |- ( 1 e. ( ZZ>= ` 2 ) -> 0 e. NN )
23		0nnn 8884	. . . . 5 |- -. 0 e. NN
24	23	pm2.21i 636	. . . 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 12043	. . . . . 6 |- ( x e. Prime -> x e. ZZ )
27		iddvds 11744	. . . . . 6 |- ( x e. ZZ -> x || x )
28	26, 27	syl 14	. . . . 5 |- ( x e. Prime -> x || x )
29		breq1 3985	. . . . . 6 |- ( p = x -> ( p || x <-> x || x ) )
30	29	rspcev 2830	. . . . 5 |- ( ( x e. Prime /\ x || x ) -> E. p e. Prime p || x )
31	28, 30	mpdan 418	. . . 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 9475	. . . . . . . . . 10 |- ( y e. ( ZZ>= ` 2 ) -> y e. ZZ )
35	34	ad2antrr 480	. . . . . . . . 9 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> y e. ZZ )
36		eluzelz 9475	. . . . . . . . . 10 |- ( z e. ( ZZ>= ` 2 ) -> z e. ZZ )
37	36	ad2antlr 481	. . . . . . . . 9 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> z e. ZZ )
38		dvdsmul1 11753	. . . . . . . . 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 12043	. . . . . . . . . 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 9319	. . . . . . . . 9 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> ( y x. z ) e. ZZ )
43		dvdstr 11768	. . . . . . . . 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 1228	. . . . . . . 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 425	. . . . . . 7 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> ( p || y -> p || ( y x. z ) ) )
46	45	reximdva 2568	. . . . . 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 12053	. 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 1343   e. wcel 2136   E.wrex 2445   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   0cc0 7753   1c1 7754   x. cmul 7758   - cmin 8069   NNcn 8857   2c2 8908   ZZcz 9191   ZZ>=cuz 9466   || cdvds 11727   Primecprime 12039

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-1o 6384  df-2o 6385  df-er 6501  df-en 6707  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-fl 10205  df-mod 10258  df-seqfrec 10381  df-exp 10455  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-dvds 11728  df-prm 12040

This theorem is referenced by:  prmdvdsfz  12071  isprm5lem  12073  rpexp  12085  pc2dvds  12261  oddprmdvds  12284  prmunb  12292  lgsne0  13579