dochnoncon - Mathbox for Norm Megill

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Theorem dochnoncon 41348

Description: Law of noncontradiction. The intersection of a subspace and its orthocomplement is the zero subspace. (Contributed by NM, 16-Apr-2014.)

**Hypotheses**
Ref	Expression
dochnoncon.h	⊢ 𝐻 = (LHyp‘𝐾)
dochnoncon.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochnoncon.s	⊢ 𝑆 = (LSubSp‘𝑈)
dochnoncon.z	⊢ 0 = (0_g‘𝑈)
dochnoncon.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
dochnoncon	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) = { 0 })

**Proof of Theorem dochnoncon**
Step	Hyp	Ref	Expression
1		eqid 2740	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
2		dochnoncon.s	. . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑈)
3	1, 2	lssss 20957	. . . . 5 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ⊆ (Base‘𝑈))
4		dochnoncon.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
5		dochnoncon.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6		dochnoncon.o	. . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
7	4, 5, 1, 6	dochocss 41323	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋)))
8	3, 7	sylan2 592	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋)))
9	8	ssrind 4265	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) ⊆ (( ⊥ ‘( ⊥ ‘𝑋)) ∩ ( ⊥ ‘𝑋)))
10		simpl 482	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		eqid 2740	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2740	. . . . . . . . . 10 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
13		eqid 2740	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
14	11, 4, 12, 5, 13	dihf11 41224	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((DIsoH‘𝐾)‘𝑊):(Base‘𝐾)–1-1→(LSubSp‘𝑈))
15	14	adantr 480	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ((DIsoH‘𝐾)‘𝑊):(Base‘𝐾)–1-1→(LSubSp‘𝑈))
16		f1f1orn 6873	. . . . . . . 8 ⊢ (((DIsoH‘𝐾)‘𝑊):(Base‘𝐾)–1-1→(LSubSp‘𝑈) → ((DIsoH‘𝐾)‘𝑊):(Base‘𝐾)–1-1-onto→ran ((DIsoH‘𝐾)‘𝑊))
17	15, 16	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ((DIsoH‘𝐾)‘𝑊):(Base‘𝐾)–1-1-onto→ran ((DIsoH‘𝐾)‘𝑊))
18	4, 12, 5, 1, 6	dochcl 41310	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
19	3, 18	sylan2 592	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
20	4, 5, 12, 13	dihrnlss 41234	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈))
21	19, 20	syldan 590	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈))
22	1, 13	lssss 20957	. . . . . . . . 9 ⊢ (( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈) → ( ⊥ ‘𝑋) ⊆ (Base‘𝑈))
23	21, 22	syl 17	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ( ⊥ ‘𝑋) ⊆ (Base‘𝑈))
24	4, 12, 5, 1, 6	dochcl 41310	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
25	23, 24	syldan 590	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
26		f1ocnvdm 7321	. . . . . . 7 ⊢ ((((DIsoH‘𝐾)‘𝑊):(Base‘𝐾)–1-1-onto→ran ((DIsoH‘𝐾)‘𝑊) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))) ∈ (Base‘𝐾))
27	17, 25, 26	syl2anc 583	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))) ∈ (Base‘𝐾))
28		hlop 39318	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
29	28	ad2antrr 725	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → 𝐾 ∈ OP)
30		eqid 2740	. . . . . . . 8 ⊢ (oc‘𝐾) = (oc‘𝐾)
31	11, 30	opoccl 39150	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ (^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))) ∈ (Base‘𝐾))
32	29, 27, 31	syl2anc 583	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))) ∈ (Base‘𝐾))
33		eqid 2740	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
34	11, 33, 4, 12	dihmeet 41300	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))) ∈ (Base‘𝐾) ∧ ((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))) ∈ (Base‘𝐾)) → (((DIsoH‘𝐾)‘𝑊)‘((^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))(meet‘𝐾)((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))))) = ((((DIsoH‘𝐾)‘𝑊)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))) ∩ (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))))))
35	10, 27, 32, 34	syl3anc 1371	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (((DIsoH‘𝐾)‘𝑊)‘((^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))(meet‘𝐾)((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))))) = ((((DIsoH‘𝐾)‘𝑊)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))) ∩ (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))))))
36		eqid 2740	. . . . . . . 8 ⊢ (0.‘𝐾) = (0.‘𝐾)
37	11, 30, 33, 36	opnoncon 39164	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ (^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))) ∈ (Base‘𝐾)) → ((^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))(meet‘𝐾)((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))))) = (0.‘𝐾))
38	29, 27, 37	syl2anc 583	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ((^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))(meet‘𝐾)((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))))) = (0.‘𝐾))
39	38	fveq2d 6924	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (((DIsoH‘𝐾)‘𝑊)‘((^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))(meet‘𝐾)((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))))) = (((DIsoH‘𝐾)‘𝑊)‘(0.‘𝐾)))
40	35, 39	eqtr3d 2782	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ((((DIsoH‘𝐾)‘𝑊)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))) ∩ (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))))) = (((DIsoH‘𝐾)‘𝑊)‘(0.‘𝐾)))
41	4, 12	dihcnvid2 41230	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (((DIsoH‘𝐾)‘𝑊)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))) = ( ⊥ ‘( ⊥ ‘𝑋)))
42	25, 41	syldan 590	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (((DIsoH‘𝐾)‘𝑊)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))) = ( ⊥ ‘( ⊥ ‘𝑋)))
43	30, 4, 12, 6	dochvalr 41314	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))))))
44	25, 43	syldan 590	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))))))
45	4, 12, 6	dochoc 41324	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑋))
46	19, 45	syldan 590	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑋))
47	44, 46	eqtr3d 2782	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))))) = ( ⊥ ‘𝑋))
48	42, 47	ineq12d 4242	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ((((DIsoH‘𝐾)‘𝑊)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))) ∩ (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))))) = (( ⊥ ‘( ⊥ ‘𝑋)) ∩ ( ⊥ ‘𝑋)))
49		dochnoncon.z	. . . . . 6 ⊢ 0 = (0_g‘𝑈)
50	36, 4, 12, 5, 49	dih0 41237	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((DIsoH‘𝐾)‘𝑊)‘(0.‘𝐾)) = { 0 })
51	50	adantr 480	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (((DIsoH‘𝐾)‘𝑊)‘(0.‘𝐾)) = { 0 })
52	40, 48, 51	3eqtr3d 2788	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (( ⊥ ‘( ⊥ ‘𝑋)) ∩ ( ⊥ ‘𝑋)) = { 0 })
53	9, 52	sseqtrd 4049	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) ⊆ { 0 })
54	4, 5, 10	dvhlmod 41067	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → 𝑈 ∈ LMod)
55		simpr 484	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
56	4, 5, 12, 2	dihrnlss 41234	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘𝑋) ∈ 𝑆)
57	19, 56	syldan 590	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ( ⊥ ‘𝑋) ∈ 𝑆)
58	2	lssincl 20986	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ ( ⊥ ‘𝑋) ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) ∈ 𝑆)
59	54, 55, 57, 58	syl3anc 1371	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) ∈ 𝑆)
60	49, 2	lss0ss 20970	. . 3 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 ∩ ( ⊥ ‘𝑋)) ∈ 𝑆) → { 0 } ⊆ (𝑋 ∩ ( ⊥ ‘𝑋)))
61	54, 59, 60	syl2anc 583	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → { 0 } ⊆ (𝑋 ∩ ( ⊥ ‘𝑋)))
62	53, 61	eqssd 4026	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) = { 0 })

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∩ cin 3975 ⊆ wss 3976 {csn 4648 ^◡ccnv 5699 ran crn 5701 –1-1→wf1 6570 –1-1-onto→wf1o 6572 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 occoc 17319 0_gc0g 17499 meetcmee 18382 0.cp0 18493 LModclmod 20880 LSubSpclss 20952 OPcops 39128 HLchlt 39306 LHypclh 39941 DVecHcdvh 41035 DIsoHcdih 41185 ocHcoch 41304

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-riotaBAD 38909

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-tpos 8267 df-undef 8314 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-0g 17501 df-proset 18365 df-poset 18383 df-plt 18400 df-lub 18416 df-glb 18417 df-join 18418 df-meet 18419 df-p0 18495 df-p1 18496 df-lat 18502 df-clat 18569 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-grp 18976 df-minusg 18977 df-sbg 18978 df-subg 19163 df-cntz 19357 df-lsm 19678 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-drng 20753 df-lmod 20882 df-lss 20953 df-lsp 20993 df-lvec 21125 df-lsatoms 38932 df-oposet 39132 df-ol 39134 df-oml 39135 df-covers 39222 df-ats 39223 df-atl 39254 df-cvlat 39278 df-hlat 39307 df-llines 39455 df-lplanes 39456 df-lvols 39457 df-lines 39458 df-psubsp 39460 df-pmap 39461 df-padd 39753 df-lhyp 39945 df-laut 39946 df-ldil 40061 df-ltrn 40062 df-trl 40116 df-tendo 40712 df-edring 40714 df-disoa 40986 df-dvech 41036 df-dib 41096 df-dic 41130 df-dih 41186 df-doch 41305

This theorem is referenced by: dochnel2 41349 djhexmid 41368 dochexmidlem1 41417 lcfrlem23 41522

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