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Theorem pridlc3 37551
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 37478 . . . 4 (𝑅 ∈ CRingOps β†’ 𝑅 ∈ RingOps)
2 eldifi 4125 . . . . 5 (𝐴 ∈ (𝑋 βˆ– 𝑃) β†’ 𝐴 ∈ 𝑋)
3 eldifi 4125 . . . . 5 (𝐡 ∈ (𝑋 βˆ– 𝑃) β†’ 𝐡 ∈ 𝑋)
42, 3anim12i 611 . . . 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 37379 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
983expb 1117 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
101, 4, 9syl2an 594 . . 3 ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
1110adantlr 713 . 2 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ (𝐴𝐻𝐡) ∈ 𝑋)
12 eldifn 4126 . . . 4 (𝐡 ∈ (𝑋 βˆ– 𝑃) β†’ Β¬ 𝐡 ∈ 𝑃)
1312ad2antll 727 . . 3 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ Β¬ 𝐡 ∈ 𝑃)
145, 6, 7pridlc2 37550 . . . . . . 7 (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ 𝑋 ∧ (𝐴𝐻𝐡) ∈ 𝑃)) β†’ 𝐡 ∈ 𝑃)
15143exp2 1351 . . . . . 6 ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) β†’ (𝐴 ∈ (𝑋 βˆ– 𝑃) β†’ (𝐡 ∈ 𝑋 β†’ ((𝐴𝐻𝐡) ∈ 𝑃 β†’ 𝐡 ∈ 𝑃))))
1615imp32 417 . . . . 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 681 . . 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 3958 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 394   = wceq 1533   ∈ wcel 2098   βˆ– cdif 3944  ran crn 5681  β€˜cfv 6551  (class class class)co 7424  1st c1st 7995  2nd c2nd 7996  RingOpscrngo 37372  CRingOpsccring 37471  PrIdlcpridl 37486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 7997  df-2nd 7998  df-grpo 30321  df-gid 30322  df-ginv 30323  df-ablo 30373  df-ass 37321  df-exid 37323  df-mgmOLD 37327  df-sgrOLD 37339  df-mndo 37345  df-rngo 37373  df-com2 37468  df-crngo 37472  df-idl 37488  df-pridl 37489  df-igen 37538
This theorem is referenced by: (None)
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