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Theorem pridlc3 36561
Description: Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
Hypotheses
Ref Expression
ispridlc.1 𝐺 = (1st β€˜π‘…)
ispridlc.2 𝐻 = (2nd β€˜π‘…)
ispridlc.3 𝑋 = ran 𝐺
Assertion
Ref Expression
pridlc3 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ (𝐴𝐻𝐡) ∈ (𝑋 βˆ– 𝑃))

Proof of Theorem pridlc3
StepHypRef Expression
1 crngorngo 36488 . . . 4 (𝑅 ∈ CRingOps β†’ 𝑅 ∈ RingOps)
2 eldifi 4091 . . . . 5 (𝐴 ∈ (𝑋 βˆ– 𝑃) β†’ 𝐴 ∈ 𝑋)
3 eldifi 4091 . . . . 5 (𝐡 ∈ (𝑋 βˆ– 𝑃) β†’ 𝐡 ∈ 𝑋)
42, 3anim12i 614 . . . 4 ((𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃)) β†’ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋))
5 ispridlc.1 . . . . . 6 𝐺 = (1st β€˜π‘…)
6 ispridlc.2 . . . . . 6 𝐻 = (2nd β€˜π‘…)
7 ispridlc.3 . . . . . 6 𝑋 = ran 𝐺
85, 6, 7rngocl 36389 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
983expb 1121 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
101, 4, 9syl2an 597 . . 3 ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
1110adantlr 714 . 2 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
12 eldifn 4092 . . . 4 (𝐡 ∈ (𝑋 βˆ– 𝑃) β†’ Β¬ 𝐡 ∈ 𝑃)
1312ad2antll 728 . . 3 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ Β¬ 𝐡 ∈ 𝑃)
145, 6, 7pridlc2 36560 . . . . . . 7 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ 𝑋 ∧ (𝐴𝐻𝐡) ∈ 𝑃)) β†’ 𝐡 ∈ 𝑃)
15143exp2 1355 . . . . . 6 ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) β†’ (𝐴 ∈ (𝑋 βˆ– 𝑃) β†’ (𝐡 ∈ 𝑋 β†’ ((𝐴𝐻𝐡) ∈ 𝑃 β†’ 𝐡 ∈ 𝑃))))
1615imp32 420 . . . . 5 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ 𝑋)) β†’ ((𝐴𝐻𝐡) ∈ 𝑃 β†’ 𝐡 ∈ 𝑃))
1716con3d 152 . . . 4 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ 𝑋)) β†’ (Β¬ 𝐡 ∈ 𝑃 β†’ Β¬ (𝐴𝐻𝐡) ∈ 𝑃))
183, 17sylanr2 682 . . 3 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ (Β¬ 𝐡 ∈ 𝑃 β†’ Β¬ (𝐴𝐻𝐡) ∈ 𝑃))
1913, 18mpd 15 . 2 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ Β¬ (𝐴𝐻𝐡) ∈ 𝑃)
2011, 19eldifd 3926 1 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ (𝐴𝐻𝐡) ∈ (𝑋 βˆ– 𝑃))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βˆ– cdif 3912  ran crn 5639  β€˜cfv 6501  (class class class)co 7362  1st c1st 7924  2nd c2nd 7925  RingOpscrngo 36382  CRingOpsccring 36481  PrIdlcpridl 36496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-grpo 29477  df-gid 29478  df-ginv 29479  df-ablo 29529  df-ass 36331  df-exid 36333  df-mgmOLD 36337  df-sgrOLD 36349  df-mndo 36355  df-rngo 36383  df-com2 36478  df-crngo 36482  df-idl 36498  df-pridl 36499  df-igen 36548
This theorem is referenced by: (None)
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