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Theorem lpival 16306
 Description: Value of the set of principal ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
Hypotheses
Ref Expression
lpival.p LPIdeal
lpival.k RSpan
lpival.b
Assertion
Ref Expression
lpival
Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem lpival
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fveq2 5720 . . . 4
2 fveq2 5720 . . . . . 6 RSpan RSpan
32fveq1d 5722 . . . . 5 RSpan RSpan
43sneqd 3819 . . . 4 RSpan RSpan
51, 4iuneq12d 4109 . . 3 RSpan RSpan
6 df-lpidl 16304 . . 3 LPIdeal RSpan
7 fvex 5734 . . . . . 6 RSpan
87rnex 5125 . . . . 5 RSpan
9 p0ex 4378 . . . . 5
108, 9unex 4699 . . . 4 RSpan
11 iunss 4124 . . . . 5 RSpan RSpan RSpan RSpan
12 fvrn0 5745 . . . . . . 7 RSpan RSpan
13 snssi 3934 . . . . . . 7 RSpan RSpan RSpan RSpan
1412, 13ax-mp 8 . . . . . 6 RSpan RSpan
1514a1i 11 . . . . 5 RSpan RSpan
1611, 15mprgbir 2768 . . . 4 RSpan RSpan
1710, 16ssexi 4340 . . 3 RSpan
185, 6, 17fvmpt 5798 . 2 LPIdeal RSpan
19 lpival.p . 2 LPIdeal
20 lpival.b . . . 4
21 iuneq1 4098 . . . 4
2220, 21ax-mp 8 . . 3
23 lpival.k . . . . . . 7 RSpan
2423fveq1i 5721 . . . . . 6 RSpan
2524sneqi 3818 . . . . 5 RSpan
2625a1i 11 . . . 4 RSpan
2726iuneq2i 4103 . . 3 RSpan
2822, 27eqtri 2455 . 2 RSpan
2918, 19, 283eqtr4g 2492 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725   cun 3310   wss 3312  c0 3620  csn 3806  ciun 4085   crn 4871  cfv 5446  cbs 13459  crg 15650  RSpancrsp 16233  LPIdealclpidl 16302 This theorem is referenced by:  islpidl  16307 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  ax-un 4693 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-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  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-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-lpidl 16304
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