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Theorem isirred 15794
 Description: An irreducible element of a ring is a non-unit that is not the product of two non-units. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
irred.1
irred.2 Unit
irred.3 Irred
irred.4
irred.5
Assertion
Ref Expression
isirred
Distinct variable groups:   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)   (,)   (,)

Proof of Theorem isirred
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvdm 5749 . . . 4 Irred Irred
2 irred.3 . . . 4 Irred
31, 2eleq2s 2527 . . 3 Irred
4 elex 2956 . . 3 Irred
53, 4syl 16 . 2
6 eldifi 3461 . . . . . 6
7 irred.4 . . . . . 6
86, 7eleq2s 2527 . . . . 5
9 irred.1 . . . . 5
108, 9syl6eleq 2525 . . . 4
1110elfvexd 5751 . . 3
13 fvex 5734 . . . . . . . 8
14 difexg 4343 . . . . . . . 8 Unit
1513, 14mp1i 12 . . . . . . 7 Unit
16 simpr 448 . . . . . . . . 9 Unit Unit
17 simpl 444 . . . . . . . . . . . . 13 Unit
1817fveq2d 5724 . . . . . . . . . . . 12 Unit
1918, 9syl6eqr 2485 . . . . . . . . . . 11 Unit
2017fveq2d 5724 . . . . . . . . . . . 12 Unit Unit Unit
21 irred.2 . . . . . . . . . . . 12 Unit
2220, 21syl6eqr 2485 . . . . . . . . . . 11 Unit Unit
2319, 22difeq12d 3458 . . . . . . . . . 10 Unit Unit
2423, 7syl6eqr 2485 . . . . . . . . 9 Unit Unit
2516, 24eqtrd 2467 . . . . . . . 8 Unit
2617fveq2d 5724 . . . . . . . . . . . . 13 Unit
27 irred.5 . . . . . . . . . . . . 13
2826, 27syl6eqr 2485 . . . . . . . . . . . 12 Unit
2928oveqd 6090 . . . . . . . . . . 11 Unit
3029neeq1d 2611 . . . . . . . . . 10 Unit
3125, 30raleqbidv 2908 . . . . . . . . 9 Unit
3225, 31raleqbidv 2908 . . . . . . . 8 Unit
3325, 32rabeqbidv 2943 . . . . . . 7 Unit
3415, 33csbied 3285 . . . . . 6 Unit
35 df-irred 15738 . . . . . 6 Irred Unit
36 fvex 5734 . . . . . . . . . 10
379, 36eqeltri 2505 . . . . . . . . 9
38 difexg 4343 . . . . . . . . 9
3937, 38ax-mp 8 . . . . . . . 8
407, 39eqeltri 2505 . . . . . . 7
4140rabex 4346 . . . . . 6
4234, 35, 41fvmpt 5798 . . . . 5 Irred
432, 42syl5eq 2479 . . . 4
4443eleq2d 2502 . . 3
45 neeq2 2607 . . . . 5
46452ralbidv 2739 . . . 4
4746elrab 3084 . . 3
4844, 47syl6bb 253 . 2
495, 12, 48pm5.21nii 343 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1652   wcel 1725   wne 2598  wral 2697  crab 2701  cvv 2948  csb 3243   cdif 3309   cdm 4870  cfv 5446  (class class class)co 6073  cbs 13459  cmulr 13520  Unitcui 15734  Irredcir 15735 This theorem is referenced by:  isnirred  15795  isirred2  15796  opprirred  15797 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-irred 15738
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