ispridl2 - Mathbox for Jeff Madsen

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Theorem ispridl2 33837

Description: A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 33869 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.)

**Hypotheses**
Ref	Expression
ispridl2.1
ispridl2.2
ispridl2.3

**Assertion**
Ref	Expression
ispridl2	$/\$ $/\$ $/\$ $\/$

Distinct variable groups:

Allowed substitution hints:

(

)

(

)

Proof of Theorem ispridl2 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ispridl2.1	. . . . . . . . . . . . . 14
2		ispridl2.3	. . . . . . . . . . . . . 14
3	1, 2	idlss 33815	. . . . . . . . . . . . 13 $/\$
4		ssralv 3666	. . . . . . . . . . . . 13 $\/$ $\/$
5	3, 4	syl 17	. . . . . . . . . . . 12 $/\$ $\/$ $\/$
6	5	adantrr 753	. . . . . . . . . . 11 $/\$ $/\$ $\/$ $\/$
7	1, 2	idlss 33815	. . . . . . . . . . . . 13 $/\$
8		ssralv 3666	. . . . . . . . . . . . . 14 $\/$ $\/$
9	8	ralimdv 2963	. . . . . . . . . . . . 13 $\/$ $\/$
10	7, 9	syl 17	. . . . . . . . . . . 12 $/\$ $\/$ $\/$
11	10	adantrl 752	. . . . . . . . . . 11 $/\$ $/\$ $\/$ $\/$
12	6, 11	syld 47	. . . . . . . . . 10 $/\$ $/\$ $\/$ $\/$
13	12	adantlr 751	. . . . . . . . 9 $/\$ $/\$ $/\$ $\/$ $\/$
14		r19.26-2 3065	. . . . . . . . . . 11 $/\$ $\/$ $/\$ $\/$
15		pm3.35 611	. . . . . . . . . . . . 13 $/\$ $\/$ $\/$
16	15	2ralimi 2953	. . . . . . . . . . . 12 $/\$ $\/$ $\/$
17		2ralor 3109	. . . . . . . . . . . . 13 $\/$ $\/$
18		dfss3 3592	. . . . . . . . . . . . . 14
19		dfss3 3592	. . . . . . . . . . . . . 14
20	18, 19	orbi12i 543	. . . . . . . . . . . . 13 $\/$ $\/$
21	17, 20	sylbb2 228	. . . . . . . . . . . 12 $\/$ $\/$
22	16, 21	syl 17	. . . . . . . . . . 11 $/\$ $\/$ $\/$
23	14, 22	sylbir 225	. . . . . . . . . 10 $/\$ $\/$ $\/$
24	23	expcom 451	. . . . . . . . 9 $\/$ $\/$
25	13, 24	syl6 35	. . . . . . . 8 $/\$ $/\$ $/\$ $\/$ $\/$
26	25	ralrimdvva 2974	. . . . . . 7 $/\$ $\/$ $\/$
27	26	ex 450	. . . . . 6 $\/$ $\/$
28	27	adantrd 484	. . . . 5 $/\$ $\/$ $\/$
29	28	imdistand 728	. . . 4 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$
30		df-3an 1039	. . . 4 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$
31		df-3an 1039	. . . 4 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$
32	29, 30, 31	3imtr4g 285	. . 3 $/\$ $/\$ $\/$ $/\$ $/\$ $\/$
33		ispridl2.2	. . . 4
34	1, 33, 2	ispridl 33833	. . 3 $/\$ $/\$ $\/$
35	32, 34	sylibrd 249	. 2 $/\$ $/\$ $\/$
36	35	imp 445	1 $/\$ $/\$ $/\$ $\/$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $\/$ wo 383 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

wral 2912

wss 3574

crn 5115

cfv 5888 (class class class)co 6650

c1st 7166

c2nd 7167

crngo 33693

cidl 33806

cpridl 33807

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-idl 33809 df-pridl 33810

This theorem is referenced by: ispridlc 33869

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