Theorem mdsl2 10249

Description: If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2.

Hypotheses

Ref	Expression
mdsl.1	|- A e. CH
mdsl.2	|- B e. CH

Assertion

Ref	Expression
mdsl2	|- (A MH B <-> A.x e. CH (((A i^i B) (_ x /\ x (_ B) -> ((x vH A) i^i B) (_ (x vH (A i^i B))))

Distinct variable groups:   x,A   x,B

Proof of Theorem mdsl2

Step	Hyp	Ref	Expression
1		iba 642	. . . . . . . . . . . 12 |- (x (_ B -> (x (_ (x vH A) <-> (x (_ (x vH A) /\ x (_ B)))
2		ssin 2232	. . . . . . . . . . . 12 |- ((x (_ (x vH A) /\ x (_ B) <-> x (_ ((x vH A) i^i B))
3	1, 2	syl6bb 536	. . . . . . . . . . 11 |- (x (_ B -> (x (_ (x vH A) <-> x (_ ((x vH A) i^i B)))
4		mdsl.1	. . . . . . . . . . . 12 |- A e. CH
5		chub1t 9430	. . . . . . . . . . . 12 |- ((x e. CH /\ A e. CH) -> x (_ (x vH A))
6	4, 5	mpan2 696	. . . . . . . . . . 11 |- (x e. CH -> x (_ (x vH A))
7	3, 6	syl5cbi 209	. . . . . . . . . 10 |- (x e. CH -> (x (_ B -> x (_ ((x vH A) i^i B)))
8		chub2t 9431	. . . . . . . . . . . 12 |- ((A e. CH /\ x e. CH) -> A (_ (x vH A))
9	4, 8	mpan 695	. . . . . . . . . . 11 |- (x e. CH -> A (_ (x vH A))
10		ssrin 2234	. . . . . . . . . . 11 |- (A (_ (x vH A) -> (A i^i B) (_ ((x vH A) i^i B))
11	9, 10	syl 10	. . . . . . . . . 10 |- (x e. CH -> (A i^i B) (_ ((x vH A) i^i B))
12	7, 11	jctird 602	. . . . . . . . 9 |- (x e. CH -> (x (_ B -> (x (_ ((x vH A) i^i B) /\ (A i^i B) (_ ((x vH A) i^i B))))
13		chjclt 9329	. . . . . . . . . . . 12 |- ((x e. CH /\ A e. CH) -> (x vH A) e. CH)
14	4, 13	mpan2 696	. . . . . . . . . . 11 |- (x e. CH -> (x vH A) e. CH)
15		mdsl.2	. . . . . . . . . . . 12 |- B e. CH
16		chinclt 9422	. . . . . . . . . . . 12 |- (((x vH A) e. CH /\ B e. CH) -> ((x vH A) i^i B) e. CH)
17	15, 16	mpan2 696	. . . . . . . . . . 11 |- ((x vH A) e. CH -> ((x vH A) i^i B) e. CH)
18	14, 17	syl 10	. . . . . . . . . 10 |- (x e. CH -> ((x vH A) i^i B) e. CH)
19	4, 15	chincl 9383	. . . . . . . . . . 11 |- (A i^i B) e. CH
20		chlubt 9432	. . . . . . . . . . 11 |- ((x e. CH /\ (A i^i B) e. CH /\ ((x vH A) i^i B) e. CH) -> ((x (_ ((x vH A) i^i B) /\ (A i^i B) (_ ((x vH A) i^i B)) <-> (x vH (A i^i B)) (_ ((x vH A) i^i B)))
21	19, 20	mp3an2 904	. . . . . . . . . 10 |- ((x e. CH /\ ((x vH A) i^i B) e. CH) -> ((x (_ ((x vH A) i^i B) /\ (A i^i B) (_ ((x vH A) i^i B)) <-> (x vH (A i^i B)) (_ ((x vH A) i^i B)))
22	18, 21	mpdan 704	. . . . . . . . 9 |- (x e. CH -> ((x (_ ((x vH A) i^i B) /\ (A i^i B) (_ ((x vH A) i^i B)) <-> (x vH (A i^i B)) (_ ((x vH A) i^i B)))
23	12, 22	sylibd 202	. . . . . . . 8 |- (x e. CH -> (x (_ B -> (x vH (A i^i B)) (_ ((x vH A) i^i B)))
24		iba 642	. . . . . . . . 9 |- ((x vH (A i^i B)) (_ ((x vH A) i^i B) -> (((x vH A) i^i B) (_ (x vH (A i^i B)) <-> (((x vH A) i^i B) (_ (x vH (A i^i B)) /\ (x vH (A i^i B)) (_ ((x vH A) i^i B))))
25		eqss 2077	. . . . . . . . 9 |- (((x vH A) i^i B) = (x vH (A i^i B)) <-> (((x vH A) i^i B) (_ (x vH (A i^i B)) /\ (x vH (A i^i B)) (_ ((x vH A) i^i B)))
26	24, 25	syl6rbbr 539	. . . . . . . 8 |- ((x vH (A i^i B)) (_ ((x vH A) i^i B) -> (((x vH A) i^i B) = (x vH (A i^i B)) <-> ((x vH A) i^i B) (_ (x vH (A i^i B))))
27	23, 26	syl6 22	. . . . . . 7 |- (x e. CH -> (x (_ B -> (((x vH A) i^i B) = (x vH (A i^i B)) <-> ((x vH A) i^i B) (_ (x vH (A i^i B)))))
28	27	adantld 390	. . . . . 6 |- (x e. CH -> (((A i^i B) (_ x /\ x (_ B) -> (((x vH A) i^i B) = (x vH (A i^i B)) <-> ((x vH A) i^i B) (_ (x vH (A i^i B)))))
29	28	pm5.74d 585	. . . . 5 |- (x e. CH -> ((((A i^i B) (_ x /\ x (_ B) -> ((x vH A) i^i B) = (x vH (A i^i B))) <-> (((A i^i B) (_ x /\ x (_ B) -> ((x vH A) i^i B) (_ (x vH (A i^i B)))))
30	15, 4	chub2 9393	. . . . . . . . . 10 |- B (_ (A vH B)
31		sstr 2072	. . . . . . . . . 10 |- ((x (_ B /\ B (_ (A vH B)) -> x (_ (A vH B))
32	30, 31	mpan2 696	. . . . . . . . 9 |- (x (_ B -> x (_ (A vH B))
33	32	pm4.71ri 638	. . . . . . . 8 |- (x (_ B <-> (x (_ (A vH B) /\ x (_ B))
34	33	anbi2i 480	. . . . . . 7 |- (((A i^i B) (_ x /\ x (_ B) <-> ((A i^i B) (_ x /\ (x (_ (A vH B) /\ x (_ B)))
35		anass 439	. . . . . . 7 |- ((((A i^i B) (_ x /\ x (_ (A vH B)) /\ x (_ B) <-> ((A i^i B) (_ x /\ (x (_ (A vH B) /\ x (_ B)))
36	34, 35	bitr4 176	. . . . . 6 |- (((A i^i B) (_ x /\ x (_ B) <-> (((A i^i B) (_ x /\ x (_ (A vH B)) /\ x (_ B))
37	36	imbi1i 186	. . . . 5 |- ((((A i^i B) (_ x /\ x (_ B) -> ((x vH A) i^i B) = (x vH (A i^i B))) <-> ((((A i^i B) (_ x /\ x (_ (A vH B)) /\ x (_ B) -> ((x vH A) i^i B) = (x vH (A i^i B))))
38	29, 37	syl5rbbr 535	. . . 4 |- (x e. CH -> ((((A i^i B) (_ x /\ x (_ B) -> ((x vH A) i^i B) (_ (x vH (A i^i B))) <-> ((((A i^i B) (_ x /\ x (_ (A vH B)) /\ x (_ B) -> ((x vH A) i^i B) = (x vH (A i^i B)))))
39		impexp 347	. . . 4 |- (((((A i^i B) (_ x /\ x (_ (A vH B)) /\ x (_ B) -> ((x vH A) i^i B) = (x vH (A i^i B))) <-> (((A i^i B) (_ x /\ x (_ (A vH B)) -> (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B)))))
40	38, 39	syl6bb 536	. . 3 |- (x e. CH -> ((((A i^i B) (_ x /\ x (_ B) -> ((x vH A) i^i B) (_ (x vH (A i^i B))) <-> (((A i^i B) (_ x /\ x (_ (A vH B)) -> (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B))))))
41	40	ralbiia 1673	. 2 |- (A.x e. CH (((A i^i B) (_ x /\ x (_ B) -> ((x vH A) i^i B) (_ (x vH (A i^i B))) <-> A.x e. CH (((A i^i B) (_ x /\ x (_ (A vH B)) -> (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B)))))
42	4, 15	mdsl1 10248	. 2 |- (A.x e. CH (((A i^i B) (_ x /\ x (_ (A vH B)) -> (x (_ B -> ((x vH A) i^i B) = (x vH (A i^i B)))) <-> A MH B)
43	41, 42	bitr2 174	1 |- (A MH B <-> A.x e. CH (((A i^i B) (_ x /\ x (_ B) -> ((x vH A) i^i B) (_ (x vH (A i^i B))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645   i^i cin 2046   (_ wss 2047   class class class wbr 2619  (class class class)co 3963  CHcch 8798   vH chj 8802   MH cmd 8835

This theorem is referenced by:  mdsl2b 10250  mdslmd1 10256

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625  ax-ac 4744  ax-hilex 8869  ax-hfvadd 8870  ax-hvcom 8871  ax-hvass 8872  ax-hv0cl 8873  ax-hvaddid 8874  ax-hfvmul 8875  ax-hvmulid 8876  ax-hvmulass 8877  ax-hvdistr1 8878  ax-hvdistr2 8879  ax-hvmul0 8880  ax-hfi 8946  ax-his1 8949  ax-his2 8950  ax-his3 8951  ax-his4 8952  ax-hcompl 9071

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-iin 2569  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190