dihmeetlem1N - Mathbox for Norm Megill

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Theorem dihmeetlem1N 36087

Description: Isomorphism H of a conjunction. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
dihglblem5a.b
dihglblem5a.m	$./\$
dihglblem5a.h
dihglblem5a.i
dihglblem5a.l
dihglblem5a.j	$.\/$
dihglblem5a.a
dihglblem5a.p
dihglblem5a.t
dihglblem5a.r
dihglblem5a.e
dihglblem5a.g
dihglblem5a.o

**Assertion**
Ref	Expression
dihmeetlem1N	$/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Distinct variable groups: $./\$ ,

Allowed substitution hints:

(

)

(

)

(

)

(

)

(

)

(

) $.\/$ (

) $./\$ (

)

(

)

(

)

(

)

Proof of Theorem dihmeetlem1N Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1l 1020	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
2		hllat 34160	. . . . . 6
3	1, 2	syl 16	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
4		simp2l 1022	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
5		simp3l 1024	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
6		dihglblem5a.b	. . . . . 6
7		dihglblem5a.l	. . . . . 6
8		dihglblem5a.m	. . . . . 6 $./\$
9	6, 7, 8	latmle1 15559	. . . . 5 $/\$ $/\$ $./\$
10	3, 4, 5, 9	syl3anc 1228	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
11		simp1 996	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12	6, 8	latmcl 15535	. . . . . 6 $/\$ $/\$ $./\$
13	3, 4, 5, 12	syl3anc 1228	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
14		dihglblem5a.h	. . . . . 6
15		dihglblem5a.i	. . . . . 6
16	6, 7, 14, 15	dihord 36061	. . . . 5 $/\$ $/\$ $./\$ $/\$ $./\$ $./\$
17	11, 13, 4, 16	syl3anc 1228	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$
18	10, 17	mpbird 232	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
19	6, 7, 8	latmle2 15560	. . . . 5 $/\$ $/\$ $./\$
20	3, 4, 5, 19	syl3anc 1228	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
21	6, 7, 14, 15	dihord 36061	. . . . 5 $/\$ $/\$ $./\$ $/\$ $./\$ $./\$
22	11, 13, 5, 21	syl3anc 1228	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$
23	20, 22	mpbird 232	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
24	18, 23	ssind 3722	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
25	14, 15	dihvalrel 36076	. . . . 5 $/\$
26		relin1 5118	. . . . 5
27	25, 26	syl 16	. . . 4 $/\$
28	27	3ad2ant1 1017	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
29		elin 3687	. . . 4 $/\$
30		dihglblem5a.j	. . . . . . 7 $.\/$
31		dihglblem5a.a	. . . . . . 7
32	6, 7, 30, 8, 31, 14	lhpmcvr2 34820	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
33	32	3adant3 1016	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
34		simpl1 999	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$
35		simpl2 1000	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$
36		simprl 755	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
37		simprrl 763	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
38	36, 37	jca 532	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$
39		simprrr 764	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
40		dihglblem5a.p	. . . . . . . . 9
41		dihglblem5a.t	. . . . . . . . 9
42		dihglblem5a.r	. . . . . . . . 9
43		dihglblem5a.e	. . . . . . . . 9
44		dihglblem5a.g	. . . . . . . . 9
45		vex 3116	. . . . . . . . 9
46		vex 3116	. . . . . . . . 9
47	6, 7, 30, 8, 31, 14, 40, 41, 42, 43, 15, 44, 45, 46	dihopelvalc 36046	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$
48	34, 35, 38, 39, 47	syl112anc 1232	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$
49		simpr 461	. . . . . . 7 $/\$ $/\$
50	48, 49	syl6bi 228	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
51		simpl3 1001	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$
52		dihglblem5a.o	. . . . . . . . 9
53	6, 7, 14, 41, 42, 52, 15	dihopelvalbN 36035	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
54	34, 51, 53	syl2anc 661	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$
55	54	biimpd 207	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$
56		simprll 761	. . . . . . . . . 10 $/\$ $/\$ $/\$
57	56	3ad2ant3 1019	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
58		simp3rr 1070	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
59	58	fveq1d 5866	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
60		simp11 1026	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$
61	7, 31, 14, 40	lhpocnel2 34815	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
62	60, 61	syl 16	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$
63		simp2l 1022	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
64		simp2rl 1065	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
65	7, 31, 14, 41, 44	ltrniotacl 35375	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$
66	60, 62, 63, 64, 65	syl112anc 1232	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
67	52, 6	tendo02 35583	. . . . . . . . . . . . . . . . . 18
68	66, 67	syl 16	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
69	59, 68	eqtrd 2508	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
70	69	cnveqd 5176	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
71		cnvresid 5656	. . . . . . . . . . . . . . 15
72	70, 71	syl6eq 2524	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
73	72	coeq2d 5163	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
74	6, 14, 41	ltrn1o 34920	. . . . . . . . . . . . . . 15 $/\$ $/\$
75	60, 57, 74	syl2anc 661	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
76		f1of 5814	. . . . . . . . . . . . . 14
77		fcoi1 5757	. . . . . . . . . . . . . 14
78	75, 76, 77	3syl 20	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
79	73, 78	eqtrd 2508	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
80	79	fveq2d 5868	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
81		simp3l 1024	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
82	80, 81	eqbrtrrd 4469	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
83		simprlr 762	. . . . . . . . . . 11 $/\$ $/\$ $/\$
84	83	3ad2ant3 1019	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
85		simp11l 1107	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
86	85, 2	syl 16	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
87	6, 14, 41, 42	trlcl 34960	. . . . . . . . . . . 12 $/\$ $/\$
88	60, 57, 87	syl2anc 661	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
89		simp12l 1109	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
90		simp13l 1111	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
91	6, 7, 8	latlem12 15561	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $./\$
92	86, 88, 89, 90, 91	syl13anc 1230	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
93	82, 84, 92	mpbi2and 919	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$
94	57, 93	jca 532	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
95	86, 89, 90, 12	syl3anc 1228	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$
96		simp11r 1108	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
97	6, 14	lhpbase 34794	. . . . . . . . . . 11
98	96, 97	syl 16	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
99	86, 89, 90, 19	syl3anc 1228	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$
100		simp13r 1112	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$
101	6, 7, 86, 95, 90, 98, 99, 100	lattrd 15541	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$
102	6, 7, 14, 41, 42, 52, 15	dihopelvalbN 36035	. . . . . . . . 9 $/\$ $/\$ $./\$ $/\$ $./\$ $./\$ $/\$ $./\$ $/\$
103	60, 95, 101, 102	syl12anc 1226	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $/\$
104	94, 58, 103	mpbir2and 920	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$
105	104	3expia 1198	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $./\$
106	50, 55, 105	syl2and 483	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $./\$
107	33, 106	rexlimddv 2959	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
108	29, 107	syl5bi 217	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
109	28, 108	relssdv 5093	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
110	24, 109	eqssd 3521	1 $/\$ $/\$ $/\$ $/\$ $/\$ $./\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 184 $/\$ wa 369 $/\$ w3a 973

wceq 1379

wcel 1767

wrex 2815

cin 3475

wss 3476

cop 4033 class class class wbr 4447

cmpt 4505

cid 4790

ccnv 4998

cres 5001

ccom 5003

wrel 5004

wf 5582

wf1o 5585

cfv 5586

crio 6242 (class class class)co 6282

cbs 14486

cple 14558

coc 14559

cjn 15427

cmee 15428

clat 15528

catm 34060

chlt 34147

clh 34780

cltrn 34897

ctrl 34954

ctendo 35548

cdih 36025

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1601 ax-4 1612 ax-5 1680 ax-6 1719 ax-7 1739 ax-8 1769 ax-9 1771 ax-10 1786 ax-11 1791 ax-12 1803 ax-13 1968 ax-ext 2445 ax-rep 4558 ax-sep 4568 ax-nul 4576 ax-pow 4625 ax-pr 4686 ax-un 6574 ax-cnex 9544 ax-resscn 9545 ax-1cn 9546 ax-icn 9547 ax-addcl 9548 ax-addrcl 9549 ax-mulcl 9550 ax-mulrcl 9551 ax-mulcom 9552 ax-addass 9553 ax-mulass 9554 ax-distr 9555 ax-i2m1 9556 ax-1ne0 9557 ax-1rid 9558 ax-rnegex 9559 ax-rrecex 9560 ax-cnre 9561 ax-pre-lttri 9562 ax-pre-lttrn 9563 ax-pre-ltadd 9564 ax-pre-mulgt0 9565 ax-riotaBAD 33756

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1382 df-fal 1385 df-ex 1597 df-nf 1600 df-sb 1712 df-eu 2279 df-mo 2280 df-clab 2453 df-cleq 2459 df-clel 2462 df-nfc 2617 df-ne 2664 df-nel 2665 df-ral 2819 df-rex 2820 df-reu 2821 df-rmo 2822 df-rab 2823 df-v 3115 df-sbc 3332 df-csb 3436 df-dif 3479 df-un 3481 df-in 3483 df-ss 3490 df-pss 3492 df-nul 3786 df-if 3940 df-pw 4012 df-sn 4028 df-pr 4030 df-tp 4032 df-op 4034 df-uni 4246 df-int 4283 df-iun 4327 df-iin 4328 df-br 4448 df-opab 4506 df-mpt 4507 df-tr 4541 df-eprel 4791 df-id 4795 df-po 4800 df-so 4801 df-fr 4838 df-we 4840 df-ord 4881 df-on 4882 df-lim 4883 df-suc 4884 df-xp 5005 df-rel 5006 df-cnv 5007 df-co 5008 df-dm 5009 df-rn 5010 df-res 5011 df-ima 5012 df-iota 5549 df-fun 5588 df-fn 5589 df-f 5590 df-f1 5591 df-fo 5592 df-f1o 5593 df-fv 5594 df-riota 6243 df-ov 6285 df-oprab 6286 df-mpt2 6287 df-om 6679 df-1st 6781 df-2nd 6782 df-tpos 6952 df-undef 6999 df-recs 7039 df-rdg 7073 df-1o 7127 df-oadd 7131 df-er 7308 df-map 7419 df-en 7514 df-dom 7515 df-sdom 7516 df-fin 7517 df-pnf 9626 df-mnf 9627 df-xr 9628 df-ltxr 9629 df-le 9630 df-sub 9803 df-neg 9804 df-nn 10533 df-2 10590 df-3 10591 df-4 10592 df-5 10593 df-6 10594 df-n0 10792 df-z 10861 df-uz 11079 df-fz 11669 df-struct 14488 df-ndx 14489 df-slot 14490 df-base 14491 df-sets 14492 df-ress 14493 df-plusg 14564 df-mulr 14565 df-sca 14567 df-vsca 14568 df-0g 14693 df-poset 15429 df-plt 15441 df-lub 15457 df-glb 15458 df-join 15459 df-meet 15460 df-p0 15522 df-p1 15523 df-lat 15529 df-clat 15591 df-mnd 15728 df-submnd 15778 df-grp 15858 df-minusg 15859 df-sbg 15860 df-subg 15993 df-cntz 16150 df-lsm 16452 df-cmn 16596 df-abl 16597 df-mgp 16932 df-ur 16944 df-rng 16988 df-oppr 17056 df-dvdsr 17074 df-unit 17075 df-invr 17105 df-dvr 17116 df-drng 17181 df-lmod 17297 df-lss 17362 df-lsp 17401 df-lvec 17532 df-oposet 33973 df-ol 33975 df-oml 33976 df-covers 34063 df-ats 34064 df-atl 34095 df-cvlat 34119 df-hlat 34148 df-llines 34294 df-lplanes 34295 df-lvols 34296 df-lines 34297 df-psubsp 34299 df-pmap 34300 df-padd 34592 df-lhyp 34784 df-laut 34785 df-ldil 34900 df-ltrn 34901 df-trl 34955 df-tendo 35551 df-edring 35553 df-disoa 35826 df-dvech 35876 df-dib 35936 df-dic 35970 df-dih 36026

This theorem is referenced by: dihmeetbN 36100

W3C validator