Theorem pridlc2 38032

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
pridlc2	⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → 𝐵 ∈ 𝑃)

Proof of Theorem pridlc2

Step	Hyp	Ref	Expression
1		eldifn 4155	. . . 4 ⊢ (𝐴 ∈ (𝑋 ∖ 𝑃) → ¬ 𝐴 ∈ 𝑃)
2	1	3ad2ant1 1133	. . 3 ⊢ ((𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃) → ¬ 𝐴 ∈ 𝑃)
3	2	adantl 481	. 2 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → ¬ 𝐴 ∈ 𝑃)
4		eldifi 4154	. . 3 ⊢ (𝐴 ∈ (𝑋 ∖ 𝑃) → 𝐴 ∈ 𝑋)
5		ispridlc.1	. . . . 5 ⊢ 𝐺 = (1st ‘𝑅)
6		ispridlc.2	. . . . 5 ⊢ 𝐻 = (2nd ‘𝑅)
7		ispridlc.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
8	5, 6, 7	pridlc 38031	. . . 4 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃))
9	8	ord 863	. . 3 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (¬ 𝐴 ∈ 𝑃 → 𝐵 ∈ 𝑃))
10	4, 9	syl3anr1 1416	. 2 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (¬ 𝐴 ∈ 𝑃 → 𝐵 ∈ 𝑃))
11	3, 10	mpd 15	1 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → 𝐵 ∈ 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ∖ cdif 3973  ran crn 5701  ‘cfv 6573  (class class class)co 7448  1^st c1st 8028  2^nd c2nd 8029  CRingOpsccring 37953  PrIdlcpridl 37968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-grpo 30525  df-gid 30526  df-ginv 30527  df-ablo 30577  df-ass 37803  df-exid 37805  df-mgmOLD 37809  df-sgrOLD 37821  df-mndo 37827  df-rngo 37855  df-com2 37950  df-crngo 37954  df-idl 37970  df-pridl 37971  df-igen 38020

This theorem is referenced by:  pridlc3  38033