Theorem pridlc3 39150

Description: Obsolete theorem, use cmprmidlmcl 21646 instead. Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
ispridlc.1	⊢ 𝐺 = (1st ‘𝑅)
ispridlc.2	⊢ 𝐻 = (2nd ‘𝑅)
ispridlc.3	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
pridlc3	⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ 𝑃))

Proof of Theorem pridlc3

Step	Hyp	Ref	Expression
1		crngorngo 39077	. . . 4 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
2		eldifi 4070	. . . . 5 ⊢ (𝐴 ∈ (𝑋 ∖ 𝑃) → 𝐴 ∈ 𝑋)
3		eldifi 4070	. . . . 5 ⊢ (𝐵 ∈ (𝑋 ∖ 𝑃) → 𝐵 ∈ 𝑋)
4	2, 3	anim12i 625	. . . 4 ⊢ ((𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃)) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
5		ispridlc.1	. . . . . 6 ⊢ 𝐺 = (1st ‘𝑅)
6		ispridlc.2	. . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅)
7		ispridlc.3	. . . . . 6 ⊢ 𝑋 = ran 𝐺
8	5, 6, 7	rngocl 38978	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
9	8	3expb 1138	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
10	1, 4, 9	syl2an 608	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → (𝐴𝐻𝐵) ∈ 𝑋)
11	10	adantlr 728	. 2 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → (𝐴𝐻𝐵) ∈ 𝑋)
12		eldifn 4071	. . . 4 ⊢ (𝐵 ∈ (𝑋 ∖ 𝑃) → ¬ 𝐵 ∈ 𝑃)
13	12	ad2antll 742	. . 3 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → ¬ 𝐵 ∈ 𝑃)
14	5, 6, 7	pridlc2 39149	. . . . . . 7 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → 𝐵 ∈ 𝑃)
15	14	3exp2 1373	. . . . . 6 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝐴 ∈ (𝑋 ∖ 𝑃) → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) ∈ 𝑃 → 𝐵 ∈ 𝑃))))
16	15	imp32 424	. . . . 5 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐻𝐵) ∈ 𝑃 → 𝐵 ∈ 𝑃))
17	16	con3d 153	. . . 4 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋)) → (¬ 𝐵 ∈ 𝑃 → ¬ (𝐴𝐻𝐵) ∈ 𝑃))
18	3, 17	sylanr2 696	. . 3 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → (¬ 𝐵 ∈ 𝑃 → ¬ (𝐴𝐻𝐵) ∈ 𝑃))
19	13, 18	mpd 16	. 2 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → ¬ (𝐴𝐻𝐵) ∈ 𝑃)
20	11, 19	eldifd 4053	1 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ 𝑃))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146   ∖ cdif 4048  ran crn 5656  ‘cfv 6547  (class class class)co 7430  1^st c1st 8009  2^nd c2nd 8010  RingOpscrngo 38971  CRingOpsccring 39070  PrIdlcpridl 39085

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8011  df-2nd 8012  df-grpo 31110  df-gid 31111  df-ginv 31112  df-ablo 31162  df-ass 38920  df-exid 38922  df-mgmOLD 38926  df-sgrOLD 38938  df-mndo 38944  df-rngo 38972  df-com2 39067  df-crngo 39071  df-idl 39087  df-pridl 39088  df-igen 39137

This theorem is used by: (None)