Theorem pridlc 38808

Description: Obsolete theorem, use prmidlc 21537 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
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 38807	. . . 4 ⊢ (𝑅 ∈ CRingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))))
5	4	biimpa 482	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ 𝑋 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
6	5	simp3d 1162	. 2 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))
7		oveq1 7423	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎𝐻𝑏) = (𝐴𝐻𝑏))
8	7	eleq1d 2847	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑎𝐻𝑏) ∈ 𝑃 ↔ (𝐴𝐻𝑏) ∈ 𝑃))
9		eleq1 2850	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ 𝑃 ↔ 𝐴 ∈ 𝑃))
10	9	orbi1d 930	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃) ↔ (𝐴 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))
11	8, 10	imbi12d 347	. . . . . 6 ⊢ (𝑎 = 𝐴 → (((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) ↔ ((𝐴𝐻𝑏) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃))))
12		oveq2 7424	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝐴𝐻𝑏) = (𝐴𝐻𝐵))
13	12	eleq1d 2847	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((𝐴𝐻𝑏) ∈ 𝑃 ↔ (𝐴𝐻𝐵) ∈ 𝑃))
14		eleq1 2850	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ 𝑃 ↔ 𝐵 ∈ 𝑃))
15	14	orbi2d 929	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((𝐴 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃) ↔ (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃)))
16	13, 15	imbi12d 347	. . . . . 6 ⊢ (𝑏 = 𝐵 → (((𝐴𝐻𝑏) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) ↔ ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))))
17	11, 16	rspc2v 3590	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))))
18	17	com12 33	. . . 4 ⊢ (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))))
19	18	expd 421	. . 3 ⊢ (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)) → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) ∈ 𝑃 → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃)))))
20	19	3imp2 1368	. 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 592	1 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  ran crn 5660  ‘cfv 6537  (class class class)co 7416  1^st c1st 7987  2^nd c2nd 7988  CRingOpsccring 38730  Idlcidl 38744  PrIdlcpridl 38745

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-grpo 30960  df-gid 30961  df-ginv 30962  df-ablo 31012  df-ass 38580  df-exid 38582  df-mgmOLD 38586  df-sgrOLD 38598  df-mndo 38604  df-rngo 38632  df-com2 38727  df-crngo 38731  df-idl 38747  df-pridl 38748  df-igen 38797

This theorem is used by:  pridlc2  38809