Theorem exprmfct 12510

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 9700	. 2 |- ( N e. ( ZZ>= ` 2 ) -> N e. NN )
2		eleq1 2269	. . . 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 2269	. . . 4 |- ( x = y -> ( x e. ( ZZ>= ` 2 ) <-> y e. ( ZZ>= ` 2 ) ) )
5		breq2 4052	. . . . 5 |- ( x = y -> ( p || x <-> p || y ) )
6	5	rexbidv 2508	. . . 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 2269	. . . 4 |- ( x = z -> ( x e. ( ZZ>= ` 2 ) <-> z e. ( ZZ>= ` 2 ) ) )
9		breq2 4052	. . . . 5 |- ( x = z -> ( p || x <-> p || z ) )
10	9	rexbidv 2508	. . . 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 2269	. . . 4 |- ( x = ( y x. z ) -> ( x e. ( ZZ>= ` 2 ) <-> ( y x. z ) e. ( ZZ>= ` 2 ) ) )
13		breq2 4052	. . . . 5 |- ( x = ( y x. z ) -> ( p || x <-> p || ( y x. z ) ) )
14	13	rexbidv 2508	. . . 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 2269	. . . 4 |- ( x = N -> ( x e. ( ZZ>= ` 2 ) <-> N e. ( ZZ>= ` 2 ) ) )
17		breq2 4052	. . . . 5 |- ( x = N -> ( p || x <-> p || N ) )
18	17	rexbidv 2508	. . . 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 9118	. . . . 5 |- ( 1 - 1 ) = 0
21		uz2m1nn 9739	. . . . 5 |- ( 1 e. ( ZZ>= ` 2 ) -> ( 1 - 1 ) e. NN )
22	20, 21	eqeltrrid 2294	. . . 4 |- ( 1 e. ( ZZ>= ` 2 ) -> 0 e. NN )
23		0nnn 9076	. . . . 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 12483	. . . . . 6 |- ( x e. Prime -> x e. ZZ )
27		iddvds 12165	. . . . . 6 |- ( x e. ZZ -> x || x )
28	26, 27	syl 14	. . . . 5 |- ( x e. Prime -> x || x )
29		breq1 4051	. . . . . 6 |- ( p = x -> ( p || x <-> x || x ) )
30	29	rspcev 2879	. . . . 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 9670	. . . . . . . . . 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 9670	. . . . . . . . . 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 12174	. . . . . . . . 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 12483	. . . . . . . . . 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 9514	. . . . . . . . 9 |- ( ( ( y e. ( ZZ>= ` 2 ) /\ z e. ( ZZ>= ` 2 ) ) /\ p e. Prime ) -> ( y x. z ) e. ZZ )
43		dvdstr 12189	. . . . . . . . 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 1250	. . . . . . . 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 2609	. . . . . 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 12493	. 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 1373   e. wcel 2177   E.wrex 2486   class class class wbr 4048   ` cfv 5277  (class class class)co 5954   0cc0 7938   1c1 7939   x. cmul 7943   - cmin 8256   NNcn 9049   2c2 9100   ZZcz 9385   ZZ>=cuz 9661   || cdvds 12148   Primecprime 12479

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4164  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-iinf 4641  ax-cnex 8029  ax-resscn 8030  ax-1cn 8031  ax-1re 8032  ax-icn 8033  ax-addcl 8034  ax-addrcl 8035  ax-mulcl 8036  ax-mulrcl 8037  ax-addcom 8038  ax-mulcom 8039  ax-addass 8040  ax-mulass 8041  ax-distr 8042  ax-i2m1 8043  ax-0lt1 8044  ax-1rid 8045  ax-0id 8046  ax-rnegex 8047  ax-precex 8048  ax-cnre 8049  ax-pre-ltirr 8050  ax-pre-ltwlin 8051  ax-pre-lttrn 8052  ax-pre-apti 8053  ax-pre-ltadd 8054  ax-pre-mulgt0 8055  ax-pre-mulext 8056  ax-arch 8057  ax-caucvg 8058

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-if 3574  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-iun 3932  df-br 4049  df-opab 4111  df-mpt 4112  df-tr 4148  df-id 4345  df-po 4348  df-iso 4349  df-iord 4418  df-on 4420  df-ilim 4421  df-suc 4423  df-iom 4644  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-riota 5909  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-recs 6401  df-frec 6487  df-1o 6512  df-2o 6513  df-er 6630  df-en 6838  df-pnf 8122  df-mnf 8123  df-xr 8124  df-ltxr 8125  df-le 8126  df-sub 8258  df-neg 8259  df-reap 8661  df-ap 8668  df-div 8759  df-inn 9050  df-2 9108  df-3 9109  df-4 9110  df-n0 9309  df-z 9386  df-uz 9662  df-q 9754  df-rp 9789  df-fz 10144  df-fzo 10278  df-fl 10426  df-mod 10481  df-seqfrec 10606  df-exp 10697  df-cj 11203  df-re 11204  df-im 11205  df-rsqrt 11359  df-abs 11360  df-dvds 12149  df-prm 12480

This theorem is referenced by:  prmdvdsfz  12511  isprm5lem  12513  rpexp  12525  pc2dvds  12703  oddprmdvds  12727  prmunb  12735  lgsne0  15565