Theorem pridlc3 38276

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
pridlc3	⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ 𝑃))

Proof of Theorem pridlc3

Step	Hyp	Ref	Expression
1		crngorngo 38203	. . . 4 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
2		eldifi 4083	. . . . 5 ⊢ (𝐴 ∈ (𝑋 ∖ 𝑃) → 𝐴 ∈ 𝑋)
3		eldifi 4083	. . . . 5 ⊢ (𝐵 ∈ (𝑋 ∖ 𝑃) → 𝐵 ∈ 𝑋)
4	2, 3	anim12i 613	. . . 4 ⊢ ((𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃)) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋))
5		ispridlc.1	. . . . . 6 ⊢ 𝐺 = (1st ‘𝑅)
6		ispridlc.2	. . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅)
7		ispridlc.3	. . . . . 6 ⊢ 𝑋 = ran 𝐺
8	5, 6, 7	rngocl 38104	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
9	8	3expb 1120	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
10	1, 4, 9	syl2an 596	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → (𝐴𝐻𝐵) ∈ 𝑋)
11	10	adantlr 715	. 2 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → (𝐴𝐻𝐵) ∈ 𝑋)
12		eldifn 4084	. . . 4 ⊢ (𝐵 ∈ (𝑋 ∖ 𝑃) → ¬ 𝐵 ∈ 𝑃)
13	12	ad2antll 729	. . 3 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → ¬ 𝐵 ∈ 𝑃)
14	5, 6, 7	pridlc2 38275	. . . . . . 7 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → 𝐵 ∈ 𝑃)
15	14	3exp2 1355	. . . . . 6 ⊢ ((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → (𝐴 ∈ (𝑋 ∖ 𝑃) → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) ∈ 𝑃 → 𝐵 ∈ 𝑃))))
16	15	imp32 418	. . . . 5 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐻𝐵) ∈ 𝑃 → 𝐵 ∈ 𝑃))
17	16	con3d 152	. . . 4 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋)) → (¬ 𝐵 ∈ 𝑃 → ¬ (𝐴𝐻𝐵) ∈ 𝑃))
18	3, 17	sylanr2 683	. . 3 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → (¬ 𝐵 ∈ 𝑃 → ¬ (𝐴𝐻𝐵) ∈ 𝑃))
19	13, 18	mpd 15	. 2 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → ¬ (𝐴𝐻𝐵) ∈ 𝑃)
20	11, 19	eldifd 3912	1 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ (𝑋 ∖ 𝑃))) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ 𝑃))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∖ cdif 3898  ran crn 5625  ‘cfv 6492  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  RingOpscrngo 38097  CRingOpsccring 38196  PrIdlcpridl 38211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-grpo 30570  df-gid 30571  df-ginv 30572  df-ablo 30622  df-ass 38046  df-exid 38048  df-mgmOLD 38052  df-sgrOLD 38064  df-mndo 38070  df-rngo 38098  df-com2 38193  df-crngo 38197  df-idl 38213  df-pridl 38214  df-igen 38263

This theorem is referenced by: (None)