Theorem pridlc 38065

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
pridlc	⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))

Proof of Theorem pridlc

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ispridlc.1	. . . . 5 ⊢ 𝐺 = (1st ‘𝑅)
2		ispridlc.2	. . . . 5 ⊢ 𝐻 = (2nd ‘𝑅)
3		ispridlc.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
4	1, 2, 3	ispridlc 38064	. . . 4 ⊢ (𝑅 ∈ CRingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))))
5	4	biimpa 476	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
6	5	simp3d 1144	. 2 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))
7		oveq1 7394	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎𝐻𝑏) = (𝐴𝐻𝑏))
8	7	eleq1d 2813	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑎𝐻𝑏) ∈ 𝑃 ↔ (𝐴𝐻𝑏) ∈ 𝑃))
9		eleq1 2816	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ 𝑃 ↔ 𝐴 ∈ 𝑃))
10	9	orbi1d 916	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃) ↔ (𝐴 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))
11	8, 10	imbi12d 344	. . . . . 6 ⊢ (𝑎 = 𝐴 → (((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) ↔ ((𝐴𝐻𝑏) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
12		oveq2 7395	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝐴𝐻𝑏) = (𝐴𝐻𝐵))
13	12	eleq1d 2813	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((𝐴𝐻𝑏) ∈ 𝑃 ↔ (𝐴𝐻𝐵) ∈ 𝑃))
14		eleq1 2816	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ 𝑃 ↔ 𝐵 ∈ 𝑃))
15	14	orbi2d 915	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃) ↔ (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃)))
16	13, 15	imbi12d 344	. . . . . 6 ⊢ (𝑏 = 𝐵 → (((𝐴𝐻𝑏) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) ↔ ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))))
17	11, 16	rspc2v 3599	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))))
18	17	com12 32	. . . 4 ⊢ (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))))
19	18	expd 415	. . 3 ⊢ (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃)))))
20	19	3imp2 1350	. 2 ⊢ ((∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))
21	6, 20	sylan 580	1 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ran crn 5639  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  CRingOpsccring 37987  Idlcidl 38001  PrIdlcpridl 38002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  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 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-grpo 30422  df-gid 30423  df-ginv 30424  df-ablo 30474  df-ass 37837  df-exid 37839  df-mgmOLD 37843  df-sgrOLD 37855  df-mndo 37861  df-rngo 37889  df-com2 37984  df-crngo 37988  df-idl 38004  df-pridl 38005  df-igen 38054

This theorem is referenced by:  pridlc2  38066