Theorem pridlc 38410

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 38409	. . . 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 1145	. 2 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) ∈ 𝑃 → (𝑎 ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))
7		oveq1 7369	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎𝐻𝑏) = (𝐴𝐻𝑏))
8	7	eleq1d 2822	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝑎𝐻𝑏) ∈ 𝑃 ↔ (𝐴𝐻𝑏) ∈ 𝑃))
9		eleq1 2825	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ 𝑃 ↔ 𝐴 ∈ 𝑃))
10	9	orbi1d 917	. . . . . . 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 7370	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝐴𝐻𝑏) = (𝐴𝐻𝐵))
13	12	eleq1d 2822	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((𝐴𝐻𝑏) ∈ 𝑃 ↔ (𝐴𝐻𝐵) ∈ 𝑃))
14		eleq1 2825	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝑏 ∈ 𝑃 ↔ 𝐵 ∈ 𝑃))
15	14	orbi2d 916	. . . . . . 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 3576	. . . . 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 1351	. 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 581	1 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ran crn 5627  ‘cfv 6494  (class class class)co 7362  1^st c1st 7935  2^nd c2nd 7936  CRingOpsccring 38332  Idlcidl 38346  PrIdlcpridl 38347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7937  df-2nd 7938  df-grpo 30583  df-gid 30584  df-ginv 30585  df-ablo 30635  df-ass 38182  df-exid 38184  df-mgmOLD 38188  df-sgrOLD 38200  df-mndo 38206  df-rngo 38234  df-com2 38329  df-crngo 38333  df-idl 38349  df-pridl 38350  df-igen 38399

This theorem is referenced by:  pridlc2  38411