Theorem pridlc 34175

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 34174	. . . 4 ⊢ (𝑅 ∈ CRingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))))
5	4	biimpa 502	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
6	5	simp3d 1138	. 2 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))
7		oveq1 6812	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎𝐻𝑏) = (𝐴𝐻𝑏))
8	7	eleq1d 2816	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑎𝐻𝑏) ∈ 𝑃 ↔ (𝐴𝐻𝑏) ∈ 𝑃))
9		eleq1 2819	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ 𝑃 ↔ 𝐴 ∈ 𝑃))
10	9	orbi1d 741	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃) ↔ (𝐴 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))
11	8, 10	imbi12d 333	. . . . . 6 ⊢ (𝑎 = 𝐴 → (((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) ↔ ((𝐴𝐻𝑏) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
12		oveq2 6813	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝐴𝐻𝑏) = (𝐴𝐻𝐵))
13	12	eleq1d 2816	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((𝐴𝐻𝑏) ∈ 𝑃 ↔ (𝐴𝐻𝐵) ∈ 𝑃))
14		eleq1 2819	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ 𝑃 ↔ 𝐵 ∈ 𝑃))
15	14	orbi2d 740	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃) ↔ (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃)))
16	13, 15	imbi12d 333	. . . . . 6 ⊢ (𝑏 = 𝐵 → (((𝐴𝐻𝑏) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) ↔ ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))))
17	11, 16	rspc2v 3453	. . . . 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 451	. . 3 ⊢ (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃)))))
20	19	3imp2 1440	. 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 489	1 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))

Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1072   = wceq 1624   ∈ wcel 2131   ≠ wne 2924  ∀wral 3042  ran crn 5259  ‘cfv 6041  (class class class)co 6805  1^st c1st 7323  2^nd c2nd 7324  CRingOpsccring 34097  Idlcidl 34111  PrIdlcpridl 34112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326  df-grpo 27648  df-gid 27649  df-ginv 27650  df-ablo 27700  df-ass 33947  df-exid 33949  df-mgmOLD 33953  df-sgrOLD 33965  df-mndo 33971  df-rngo 33999  df-com2 34094  df-crngo 34098  df-idl 34114  df-pridl 34115  df-igen 34164

This theorem is referenced by:  pridlc2  34176