Theorem pridlc 33870

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 33869	. . . 4 ⊢ (𝑅 ∈ CRingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))))
5	4	biimpa 501	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
6	5	simp3d 1075	. 2 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))
7		oveq1 6657	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎𝐻𝑏) = (𝐴𝐻𝑏))
8	7	eleq1d 2686	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑎𝐻𝑏) ∈ 𝑃 ↔ (𝐴𝐻𝑏) ∈ 𝑃))
9		eleq1 2689	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ 𝑃 ↔ 𝐴 ∈ 𝑃))
10	9	orbi1d 739	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃) ↔ (𝐴 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))
11	8, 10	imbi12d 334	. . . . . 6 ⊢ (𝑎 = 𝐴 → (((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) ↔ ((𝐴𝐻𝑏) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
12		oveq2 6658	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝐴𝐻𝑏) = (𝐴𝐻𝐵))
13	12	eleq1d 2686	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((𝐴𝐻𝑏) ∈ 𝑃 ↔ (𝐴𝐻𝐵) ∈ 𝑃))
14		eleq1 2689	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ 𝑃 ↔ 𝐵 ∈ 𝑃))
15	14	orbi2d 738	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃) ↔ (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃)))
16	13, 15	imbi12d 334	. . . . . 6 ⊢ (𝑏 = 𝐵 → (((𝐴𝐻𝑏) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) ↔ ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))))
17	11, 16	rspc2v 3322	. . . . 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 452	. . 3 ⊢ (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃)))))
20	19	3imp2 1282	. 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 488	1 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))

Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ran crn 5115  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  CRingOpsccring 33792  Idlcidl 33806  PrIdlcpridl 33807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694  df-com2 33789  df-crngo 33793  df-idl 33809  df-pridl 33810  df-igen 33859

This theorem is referenced by:  pridlc2  33871