Theorem mdsl2 10205

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 641	. . . . . . . . . . . 12 |- (x (_ B -> (x (_ (x vH A) <-> (x (_ (x vH A) /\ x (_ B)))
2		ssin 2229	. . . . . . . . . . . 12 |- ((x (_ (x vH A) /\ x (_ B) <-> x (_ ((x vH A) i^i B))
3	1, 2	syl6bb 535	. . . . . . . . . . 11 |- (x (_ B -> (x (_ (x vH A) <-> x (_ ((x vH A) i^i B)))
4		mdsl.1	. . . . . . . . . . . 12 |- A e. CH
5		chub1t 9385	. . . . . . . . . . . 12 |- ((x e. CH /\ A e. CH) -> x (_ (x vH A))
6	4, 5	mpan2 695	. . . . . . . . . . 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 9386	. . . . . . . . . . . 12 |- ((A e. CH /\ x e. CH) -> A (_ (x vH A))
9	4, 8	mpan 694	. . . . . . . . . . 11 |- (x e. CH -> A (_ (x vH A))
10		ssrin 2231	. . . . . . . . . . 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 601	. . . . . . . . 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 9284	. . . . . . . . . . . 12 |- ((x e. CH /\ A e. CH) -> (x vH A) e. CH)
14	4, 13	mpan2 695	. . . . . . . . . . 11 |- (x e. CH -> (x vH A) e. CH)
15		mdsl.2	. . . . . . . . . . . 12 |- B e. CH
16		chinclt 9377	. . . . . . . . . . . 12 |- (((x vH A) e. CH /\ B e. CH) -> ((x vH A) i^i B) e. CH)
17	15, 16	mpan2 695	. . . . . . . . . . 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 9338	. . . . . . . . . . 11 |- (A i^i B) e. CH
20		chlubt 9387	. . . . . . . . . . 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 903	. . . . . . . . . 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 703	. . . . . . . . 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 641	. . . . . . . . 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 2074	. . . . . . . . 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 538	. . . . . . . 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 584	. . . . 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 9348	. . . . . . . . . 10 |- B (_ (A vH B)
31		sstr 2069	. . . . . . . . . 10 |- ((x (_ B /\ B (_ (A vH B)) -> x (_ (A vH B))
32	30, 31	mpan2 695	. . . . . . . . 9 |- (x (_ B -> x (_ (A vH B))
33	32	pm4.71ri 637	. . . . . . . 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 534	. . . 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 535	. . 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 1671	. 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 10204	. 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 955   e. wcel 957  A.wral 1643   i^i cin 2043   (_ wss 2044   class class class wbr 2615  (class class class)co 3958  CHcch 8753   vH chj 8757   MH cmd 8790

This theorem is referenced by:  mdsl2b 10206  mdslmd1 10212

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-reg 4576  ax-inf2 4608  ax-ac 4727  ax-hilex 8824  ax-hfvadd 8825  ax-hvcom 8826  ax-hvass 8827  ax-hv0cl 8828  ax-hvaddid 8829  ax-hfvmul 8830  ax-hvmulid 8831  ax-hvmulass 8832  ax-hvdistr1 8833  ax-hvdistr2 8834  ax-hvmul0 8835  ax-hfi 8901  ax-his1 8904  ax-his2 8905  ax-his3 8906  ax-his4 8907  ax-hcompl 9026

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-nel 1586  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-csb 1999  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-pss 2052  df-nul 2278  df-if 2359  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-iin 2565  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-lim 2949  df-suc 2950  df-om 3128  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 31