Theorem exprmfct 12715

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 9800	. 2 |- ( N e. ( ZZ>= ` 2 ) -> N e. NN )
2		eleq1 2294	. . . 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 2294	. . . 4 |- ( x = y -> ( x e. ( ZZ>= ` 2 ) <-> y e. ( ZZ>= ` 2 ) ) )
5		breq2 4092	. . . . 5 |- ( x = y -> ( p || x <-> p || y ) )
6	5	rexbidv 2533	. . . 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 2294	. . . 4 |- ( x = z -> ( x e. ( ZZ>= ` 2 ) <-> z e. ( ZZ>= ` 2 ) ) )
9		breq2 4092	. . . . 5 |- ( x = z -> ( p || x <-> p || z ) )
10	9	rexbidv 2533	. . . 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 2294	. . . 4 |- ( x = ( y x. z ) -> ( x e. ( ZZ>= ` 2 ) <-> ( y x. z ) e. ( ZZ>= ` 2 ) ) )
13		breq2 4092	. . . . 5 |- ( x = ( y x. z ) -> ( p || x <-> p || ( y x. z ) ) )
14	13	rexbidv 2533	. . . 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 2294	. . . 4 |- ( x = N -> ( x e. ( ZZ>= ` 2 ) <-> N e. ( ZZ>= ` 2 ) ) )
17		breq2 4092	. . . . 5 |- ( x = N -> ( p || x <-> p || N ) )
18	17	rexbidv 2533	. . . 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 9212	. . . . 5 |- ( 1 - 1 ) = 0
21		uz2m1nn 9839	. . . . 5 |- ( 1 e. ( ZZ>= ` 2 ) -> ( 1 - 1 ) e. NN )
22	20, 21	eqeltrrid 2319	. . . 4 |- ( 1 e. ( ZZ>= ` 2 ) -> 0 e. NN )
23		0nnn 9170	. . . . 5 |- -. 0 e. NN
24	23	pm2.21i 651	. . . 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 12688	. . . . . 6 |- ( x e. Prime -> x e. ZZ )
27		iddvds 12370	. . . . . 6 |- ( x e. ZZ -> x || x )
28	26, 27	syl 14	. . . . 5 |- ( x e. Prime -> x || x )
29		breq1 4091	. . . . . 6 |- ( p = x -> ( p || x <-> x || x ) )
30	29	rspcev 2910	. . . . 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 9765	. . . . . . . . . 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 9765	. . . . . . . . . 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 12379	. . . . . . . . 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 12688	. . . . . . . . . 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 9608	. . . . . . . . 9 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> ( y x. z ) e. ZZ )
43		dvdstr 12394	. . . . . . . . 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 1273	. . . . . . . 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 2634	. . . . . 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 12698	. 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 1397   e. wcel 2202   E.wrex 2511   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   0cc0 8032   1c1 8033   x. cmul 8037   - cmin 8350   NNcn 9143   2c2 9194   ZZcz 9479   ZZ>=cuz 9755   || cdvds 12353   Primecprime 12684

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-1o 6582  df-2o 6583  df-er 6702  df-en 6910  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-fz 10244  df-fzo 10378  df-fl 10531  df-mod 10586  df-seqfrec 10711  df-exp 10802  df-cj 11407  df-re 11408  df-im 11409  df-rsqrt 11563  df-abs 11564  df-dvds 12354  df-prm 12685

This theorem is referenced by:  prmdvdsfz  12716  isprm5lem  12718  rpexp  12730  pc2dvds  12908  oddprmdvds  12932  prmunb  12940  lgsne0  15773