Theorem mdsl1i 28022

Description: If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
mdsl1i	|- ( A. x e. CH ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) <-> A MH B )

Distinct variable groups:   x, A   x, B

Proof of Theorem mdsl1i

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq2 3465	. . . . . . . 8 |- ( x = ( y vH ( A i^i B ) ) -> ( ( A i^i B ) C_ x <-> ( A i^i B ) C_ ( y vH ( A i^i B ) ) ) )
2		sseq1 3464	. . . . . . . 8 |- ( x = ( y vH ( A i^i B ) ) -> ( x C_ ( A vH B ) <-> ( y vH ( A i^i B ) ) C_ ( A vH B ) ) )
3	1, 2	anbi12d 722	. . . . . . 7 |- ( x = ( y vH ( A i^i B ) ) -> ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) <-> ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) ) )
4		sseq1 3464	. . . . . . . 8 |- ( x = ( y vH ( A i^i B ) ) -> ( x C_ B <-> ( y vH ( A i^i B ) ) C_ B ) )
5		oveq1 6321	. . . . . . . . . 10 |- ( x = ( y vH ( A i^i B ) ) -> ( x vH A ) = ( ( y vH ( A i^i B ) ) vH A ) )
6	5	ineq1d 3644	. . . . . . . . 9 |- ( x = ( y vH ( A i^i B ) ) -> ( ( x vH A ) i^i B ) = ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) )
7		oveq1 6321	. . . . . . . . 9 |- ( x = ( y vH ( A i^i B ) ) -> ( x vH ( A i^i B ) ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) )
8	6, 7	eqeq12d 2476	. . . . . . . 8 |- ( x = ( y vH ( A i^i B ) ) -> ( ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) <-> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) )
9	4, 8	imbi12d 326	. . . . . . 7 |- ( x = ( y vH ( A i^i B ) ) -> ( ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) <-> ( ( y vH ( A i^i B ) ) C_ B -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) ) )
10	3, 9	imbi12d 326	. . . . . 6 |- ( x = ( y vH ( A i^i B ) ) -> ( ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) <-> ( ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) -> ( ( y vH ( A i^i B ) ) C_ B -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) ) ) )
11	10	rspccv 3158	. . . . 5 |- ( A. x e. CH ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) -> ( ( y vH ( A i^i B ) ) e. CH -> ( ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) -> ( ( y vH ( A i^i B ) ) C_ B -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) ) ) )
12		impexp 452	. . . . . . 7 |- ( ( ( ( ( y vH ( A i^i B ) ) e. CH /\ ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) ) /\ ( y vH ( A i^i B ) ) C_ B ) -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) <-> ( ( ( y vH ( A i^i B ) ) e. CH /\ ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) ) -> ( ( y vH ( A i^i B ) ) C_ B -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) ) )
13		impexp 452	. . . . . . 7 |- ( ( ( ( y vH ( A i^i B ) ) e. CH /\ ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) ) -> ( ( y vH ( A i^i B ) ) C_ B -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) ) <-> ( ( y vH ( A i^i B ) ) e. CH -> ( ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) -> ( ( y vH ( A i^i B ) ) C_ B -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) ) ) )
14	12, 13	bitr2i 258	. . . . . 6 |- ( ( ( y vH ( A i^i B ) ) e. CH -> ( ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) -> ( ( y vH ( A i^i B ) ) C_ B -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) ) ) <-> ( ( ( ( y vH ( A i^i B ) ) e. CH /\ ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) ) /\ ( y vH ( A i^i B ) ) C_ B ) -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) )
15		inss2 3664	. . . . . . . . . . . 12 |- ( A i^i B ) C_ B
16		mdsl.1	. . . . . . . . . . . . . . 15 |- A e. CH
17		mdsl.2	. . . . . . . . . . . . . . 15 |- B e. CH
18	16, 17	chincli 27161	. . . . . . . . . . . . . 14 |- ( A i^i B ) e. CH
19		chlub 27210	. . . . . . . . . . . . . 14 |- ( ( y e. CH /\ ( A i^i B ) e. CH /\ B e. CH ) -> ( ( y C_ B /\ ( A i^i B ) C_ B ) <-> ( y vH ( A i^i B ) ) C_ B ) )
20	18, 17, 19	mp3an23 1365	. . . . . . . . . . . . 13 |- ( y e. CH -> ( ( y C_ B /\ ( A i^i B ) C_ B ) <-> ( y vH ( A i^i B ) ) C_ B ) )
21	20	biimpd 212	. . . . . . . . . . . 12 |- ( y e. CH -> ( ( y C_ B /\ ( A i^i B ) C_ B ) -> ( y vH ( A i^i B ) ) C_ B ) )
22	15, 21	mpan2i 688	. . . . . . . . . . 11 |- ( y e. CH -> ( y C_ B -> ( y vH ( A i^i B ) ) C_ B ) )
23	17, 16	chub2i 27171	. . . . . . . . . . . 12 |- B C_ ( A vH B )
24		sstr 3451	. . . . . . . . . . . 12 |- ( ( ( y vH ( A i^i B ) ) C_ B /\ B C_ ( A vH B ) ) -> ( y vH ( A i^i B ) ) C_ ( A vH B ) )
25	23, 24	mpan2 682	. . . . . . . . . . 11 |- ( ( y vH ( A i^i B ) ) C_ B -> ( y vH ( A i^i B ) ) C_ ( A vH B ) )
26	22, 25	syl6 34	. . . . . . . . . 10 |- ( y e. CH -> ( y C_ B -> ( y vH ( A i^i B ) ) C_ ( A vH B ) ) )
27		chub2 27209	. . . . . . . . . . 11 |- ( ( ( A i^i B ) e. CH /\ y e. CH ) -> ( A i^i B ) C_ ( y vH ( A i^i B ) ) )
28	18, 27	mpan 681	. . . . . . . . . 10 |- ( y e. CH -> ( A i^i B ) C_ ( y vH ( A i^i B ) ) )
29	26, 28	jctild 550	. . . . . . . . 9 |- ( y e. CH -> ( y C_ B -> ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) ) )
30		chjcl 27058	. . . . . . . . . 10 |- ( ( y e. CH /\ ( A i^i B ) e. CH ) -> ( y vH ( A i^i B ) ) e. CH )
31	18, 30	mpan2 682	. . . . . . . . 9 |- ( y e. CH -> ( y vH ( A i^i B ) ) e. CH )
32	29, 31	jctild 550	. . . . . . . 8 |- ( y e. CH -> ( y C_ B -> ( ( y vH ( A i^i B ) ) e. CH /\ ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) ) ) )
33	32, 22	jcad 540	. . . . . . 7 |- ( y e. CH -> ( y C_ B -> ( ( ( y vH ( A i^i B ) ) e. CH /\ ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) ) /\ ( y vH ( A i^i B ) ) C_ B ) ) )
34		chjass 27234	. . . . . . . . . . . 12 |- ( ( y e. CH /\ ( A i^i B ) e. CH /\ A e. CH ) -> ( ( y vH ( A i^i B ) ) vH A ) = ( y vH ( ( A i^i B ) vH A ) ) )
35	18, 16, 34	mp3an23 1365	. . . . . . . . . . 11 |- ( y e. CH -> ( ( y vH ( A i^i B ) ) vH A ) = ( y vH ( ( A i^i B ) vH A ) ) )
36	18, 16	chjcomi 27169	. . . . . . . . . . . . 13 |- ( ( A i^i B ) vH A ) = ( A vH ( A i^i B ) )
37	16, 17	chabs1i 27219	. . . . . . . . . . . . 13 |- ( A vH ( A i^i B ) ) = A
38	36, 37	eqtri 2483	. . . . . . . . . . . 12 |- ( ( A i^i B ) vH A ) = A
39	38	oveq2i 6325	. . . . . . . . . . 11 |- ( y vH ( ( A i^i B ) vH A ) ) = ( y vH A )
40	35, 39	syl6eq 2511	. . . . . . . . . 10 |- ( y e. CH -> ( ( y vH ( A i^i B ) ) vH A ) = ( y vH A ) )
41	40	ineq1d 3644	. . . . . . . . 9 |- ( y e. CH -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH A ) i^i B ) )
42		chjass 27234	. . . . . . . . . . 11 |- ( ( y e. CH /\ ( A i^i B ) e. CH /\ ( A i^i B ) e. CH ) -> ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) = ( y vH ( ( A i^i B ) vH ( A i^i B ) ) ) )
43	18, 18, 42	mp3an23 1365	. . . . . . . . . 10 |- ( y e. CH -> ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) = ( y vH ( ( A i^i B ) vH ( A i^i B ) ) ) )
44	18	chjidmi 27222	. . . . . . . . . . 11 |- ( ( A i^i B ) vH ( A i^i B ) ) = ( A i^i B )
45	44	oveq2i 6325	. . . . . . . . . 10 |- ( y vH ( ( A i^i B ) vH ( A i^i B ) ) ) = ( y vH ( A i^i B ) )
46	43, 45	syl6eq 2511	. . . . . . . . 9 |- ( y e. CH -> ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) = ( y vH ( A i^i B ) ) )
47	41, 46	eqeq12d 2476	. . . . . . . 8 |- ( y e. CH -> ( ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) <-> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) )
48	47	biimpd 212	. . . . . . 7 |- ( y e. CH -> ( ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) )
49	33, 48	imim12d 77	. . . . . 6 |- ( y e. CH -> ( ( ( ( ( y vH ( A i^i B ) ) e. CH /\ ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) ) /\ ( y vH ( A i^i B ) ) C_ B ) -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) -> ( y C_ B -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) ) )
50	14, 49	syl5bi 225	. . . . 5 |- ( y e. CH -> ( ( ( y vH ( A i^i B ) ) e. CH -> ( ( ( A i^i B ) C_ ( y vH ( A i^i B ) ) /\ ( y vH ( A i^i B ) ) C_ ( A vH B ) ) -> ( ( y vH ( A i^i B ) ) C_ B -> ( ( ( y vH ( A i^i B ) ) vH A ) i^i B ) = ( ( y vH ( A i^i B ) ) vH ( A i^i B ) ) ) ) ) -> ( y C_ B -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) ) )
51	11, 50	syl5com 31	. . . 4 |- ( A. x e. CH ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) -> ( y e. CH -> ( y C_ B -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) ) )
52	51	ralrimiv 2811	. . 3 |- ( A. x e. CH ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) -> A. y e. CH ( y C_ B -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) )
53		mdbr 27995	. . . 4 |- ( ( A e. CH /\ B e. CH ) -> ( A MH B <-> A. y e. CH ( y C_ B -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) ) )
54	16, 17, 53	mp2an 683	. . 3 |- ( A MH B <-> A. y e. CH ( y C_ B -> ( ( y vH A ) i^i B ) = ( y vH ( A i^i B ) ) ) )
55	52, 54	sylibr 217	. 2 |- ( A. x e. CH ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) -> A MH B )
56		mdbr 27995	. . . 4 |- ( ( A e. CH /\ B e. CH ) -> ( A MH B <-> A. x e. CH ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) )
57	16, 17, 56	mp2an 683	. . 3 |- ( A MH B <-> A. x e. CH ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) )
58		ax-1 6	. . . 4 |- ( ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) -> ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) )
59	58	ralimi 2792	. . 3 |- ( A. x e. CH ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) -> A. x e. CH ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) )
60	57, 59	sylbi 200	. 2 |- ( A MH B -> A. x e. CH ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) )
61	55, 60	impbii 192	1 |- ( A. x e. CH ( ( ( A i^i B ) C_ x /\ x C_ ( A vH B ) ) -> ( x C_ B -> ( ( x vH A ) i^i B ) = ( x vH ( A i^i B ) ) ) ) <-> A MH B )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1454   e. wcel 1897   A.wral 2748   i^i cin 3414   C_ wss 3415   class class class wbr 4415  (class class class)co 6314   CHcch 26630   vH chj 26634   MH cmd 26667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-inf2 8171  ax-cc 8890  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641  ax-pre-sup 9642  ax-addf 9643  ax-mulf 9644  ax-hilex 26700  ax-hfvadd 26701  ax-hvcom 26702  ax-hvass 26703  ax-hv0cl 26704  ax-hvaddid 26705  ax-hfvmul 26706  ax-hvmulid 26707  ax-hvmulass 26708  ax-hvdistr1 26709  ax-hvdistr2 26710  ax-hvmul0 26711  ax-hfi 26780  ax-his1 26783  ax-his2 26784  ax-his3 26785  ax-his4 26786  ax-hcompl 26903

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-iin 4294  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-se 4812  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-isom 5609  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-of 6557  df-om 6719  df-1st 6819  df-2nd 6820  df-supp 6941  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-2o 7208  df-oadd 7211  df-omul 7212  df-er 7388  df-map 7499  df-pm 7500  df-ixp 7548  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-fsupp 7909  df-fi 7950  df-sup 7981  df-inf 7982  df-oi 8050  df-card 8398  df-acn 8401  df-cda 8623  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-div 10297  df-nn 10637  df-2 10695  df-3 10696  df-4 10697  df-5 10698  df-6 10699  df-7 10700  df-8 10701  df-9 10702  df-10 10703  df-n0 10898  df-z 10966  df-dec 11080  df-uz 11188  df-q 11293  df-rp 11331  df-xneg 11437  df-xadd 11438  df-xmul 11439  df-ioo 11667  df-ico 11669  df-icc 11670  df-fz 11813  df-fzo 11946  df-fl 12059  df-seq 12245  df-exp 12304  df-hash 12547  df-cj 13210  df-re 13211  df-im 13212  df-sqrt 13346  df-abs 13347  df-clim 13600  df-rlim 13601  df-sum 13801  df-struct 15171  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-mulr 15252  df-starv 15253  df-sca 15254  df-vsca 15255  df-ip 15256  df-tset 15257  df-ple 15258  df-ds 15260  df-unif 15261  df-hom 15262  df-cco 15263  df-rest 15369  df-topn 15370  df-0g 15388  df-gsum 15389  df-topgen 15390  df-pt 15391  df-prds 15394  df-xrs 15448  df-qtop 15454  df-imas 15455  df-xps 15458  df-mre 15540  df-mrc 15541  df-acs 15543  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-submnd 16631  df-mulg 16724  df-cntz 17019  df-cmn 17480  df-psmet 19010  df-xmet 19011  df-met 19012  df-bl 19013  df-mopn 19014  df-fbas 19015  df-fg 19016  df-cnfld 19019  df-top 19969  df-bases 19970  df-topon 19971  df-topsp 19972  df-cld 20082  df-ntr 20083  df-cls 20084  df-nei 20162  df-cn 20291  df-cnp 20292  df-lm 20293  df-haus 20379  df-tx 20625  df-hmeo 20818  df-fil 20909  df-fm 21001  df-flim 21002  df-flf 21003  df-xms 21383  df-ms 21384  df-tms 21385  df-cfil 22273  df-cau 22274  df-cmet 22275  df-grpo 25967  df-gid 25968  df-ginv 25969  df-gdiv 25970  df-ablo 26058  df-subgo 26078  df-vc 26213  df-nv 26259  df-va 26262  df-ba 26263  df-sm 26264  df-0v 26265  df-vs 26266  df-nmcv 26267  df-ims 26268  df-dip 26385  df-ssp 26409  df-ph 26502  df-cbn 26553  df-hnorm 26669  df-hba 26670  df-hvsub 26672  df-hlim 26673  df-hcau 26674  df-sh 26908  df-ch 26922  df-oc 26953  df-ch0 26954  df-shs 27009  df-chj 27011  df-md 27981

This theorem is referenced by:  mdsl2i  28023  cvmdi  28025