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Mirrors > Home > MPE Home > Th. List > Mathboxes > pridlc2 | Structured version Visualization version GIF version |
Description: Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.) |
Ref | Expression |
---|---|
ispridlc.1 | ⊢ 𝐺 = (1st ‘𝑅) |
ispridlc.2 | ⊢ 𝐻 = (2nd ‘𝑅) |
ispridlc.3 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
pridlc2 | ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → 𝐵 ∈ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifn 4104 | . . . 4 ⊢ (𝐴 ∈ (𝑋 ∖ 𝑃) → ¬ 𝐴 ∈ 𝑃) | |
2 | 1 | 3ad2ant1 1129 | . . 3 ⊢ ((𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃) → ¬ 𝐴 ∈ 𝑃) |
3 | 2 | adantl 484 | . 2 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ (𝑋 ∖ 𝑃) ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → ¬ 𝐴 ∈ 𝑃) |
4 | eldifi 4103 | . . 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 35343 | . . . 4 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (𝐴 ∈ 𝑃 ∨ 𝐵 ∈ 𝑃)) |
9 | 8 | ord 860 | . . 3 ⊢ (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) ∈ 𝑃)) → (¬ 𝐴 ∈ 𝑃 → 𝐵 ∈ 𝑃)) |
10 | 4, 9 | syl3anr1 1412 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∖ cdif 3933 ran crn 5551 ‘cfv 6350 (class class class)co 7150 1st c1st 7681 2nd c2nd 7682 CRingOpsccring 35265 PrIdlcpridl 35280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-grpo 28264 df-gid 28265 df-ginv 28266 df-ablo 28316 df-ass 35115 df-exid 35117 df-mgmOLD 35121 df-sgrOLD 35133 df-mndo 35139 df-rngo 35167 df-com2 35262 df-crngo 35266 df-idl 35282 df-pridl 35283 df-igen 35332 |
This theorem is referenced by: pridlc3 35345 |
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