Theorem pridlc3 38059

Description: Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)

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 37986	. . . 4 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
2		eldifi 4140	. . . . 5 ⊢ (𝐴 ∈ (𝑋 ∖ 𝑃) → 𝐴 ∈ 𝑋)
3		eldifi 4140	. . . . 5 ⊢ (𝐵 ∈ (𝑋 ∖ 𝑃) → 𝐵 ∈ 𝑋)
4	2, 3	anim12i 613	. . . 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 37887	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
9	8	3expb 1119	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
10	1, 4, 9	syl2an 596	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → (𝐴𝐻𝐵) ∈ 𝑋)
11	10	adantlr 715	. 2 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → (𝐴𝐻𝐵) ∈ 𝑋)
12		eldifn 4141	. . . 4 ⊢ (𝐵 ∈ (𝑋 ∖ 𝑃) → ¬ 𝐵 ∈ 𝑃)
13	12	ad2antll 729	. . 3 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → ¬ 𝐵 ∈ 𝑃)
14	5, 6, 7	pridlc2 38058	. . . . . . 7 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → 𝐵 ∈ 𝑃)
15	14	3exp2 1353	. . . . . 6 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝐴 ∈ (𝑋 ∖ 𝑃) → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) ∈ 𝑃 → 𝐵 ∈ 𝑃))))
16	15	imp32 418	. . . . 5 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐻𝐵) ∈ 𝑃 → 𝐵 ∈ 𝑃))
17	16	con3d 152	. . . 4 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋)) → (¬ 𝐵 ∈ 𝑃 → ¬ (𝐴𝐻𝐵) ∈ 𝑃))
18	3, 17	sylanr2 683	. . 3 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → (¬ 𝐵 ∈ 𝑃 → ¬ (𝐴𝐻𝐵) ∈ 𝑃))
19	13, 18	mpd 15	. 2 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → ¬ (𝐴𝐻𝐵) ∈ 𝑃)
20	11, 19	eldifd 3973	1 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ 𝑃))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105   ∖ cdif 3959  ran crn 5689  ‘cfv 6562  (class class class)co 7430  1^st c1st 8010  2^nd c2nd 8011  RingOpscrngo 37880  CRingOpsccring 37979  PrIdlcpridl 37994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-grpo 30521  df-gid 30522  df-ginv 30523  df-ablo 30573  df-ass 37829  df-exid 37831  df-mgmOLD 37835  df-sgrOLD 37847  df-mndo 37853  df-rngo 37881  df-com2 37976  df-crngo 37980  df-idl 37996  df-pridl 37997  df-igen 38046

This theorem is referenced by: (None)