dochnoncon - Mathbox for Norm Megill

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Theorem dochnoncon 40566

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 2731	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
2		dochnoncon.s	. . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑈)
3	1, 2	lssss 20692	. . . . 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 40541	. . . . 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 4236	. . 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 2731	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2731	. . . . . . . . . 10 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
13		eqid 2731	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
14	11, 4, 12, 5, 13	dihf11 40442	. . . . . . . . 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 6845	. . . . . . . 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 40528	. . . . . . . . . . 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 40452	. . . . . . . . . 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 20692	. . . . . . . . 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 40528	. . . . . . . 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 7286	. . . . . . 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 38536	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
29	28	ad2antrr 723	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → 𝐾 ∈ OP)
30		eqid 2731	. . . . . . . 8 ⊢ (oc‘𝐾) = (oc‘𝐾)
31	11, 30	opoccl 38368	. . . . . . 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 2731	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
34	11, 33, 4, 12	dihmeet 40518	. . . . . 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 1370	. . . . 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 2731	. . . . . . . 8 ⊢ (0.‘𝐾) = (0.‘𝐾)
37	11, 30, 33, 36	opnoncon 38382	. . . . . . 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 6896	. . . . 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 2773	. . . 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 40448	. . . . . 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 40532	. . . . . . 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 40542	. . . . . . 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 2773	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))))) = ( ⊥ ‘𝑋))
48	42, 47	ineq12d 4214	. . . 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 40455	. . . . 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 2779	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (( ⊥ ‘( ⊥ ‘𝑋)) ∩ ( ⊥ ‘𝑋)) = { 0 })
53	9, 52	sseqtrd 4023	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) ⊆ { 0 })
54	4, 5, 10	dvhlmod 40285	. . 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 40452	. . . . 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 20721	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ ( ⊥ ‘𝑋) ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) ∈ 𝑆)
59	54, 55, 57, 58	syl3anc 1370	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) ∈ 𝑆)
60	49, 2	lss0ss 20704	. . 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 4000	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) = { 0 })

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∩ cin 3948 ⊆ wss 3949 {csn 4629 ^◡ccnv 5676 ran crn 5678 –1-1→wf1 6541 –1-1-onto→wf1o 6543 ‘cfv 6544 (class class class)co 7412 Basecbs 17149 occoc 17210 0_gc0g 17390 meetcmee 18270 0.cp0 18381 LModclmod 20615 LSubSpclss 20687 OPcops 38346 HLchlt 38524 LHypclh 39159 DVecHcdvh 40253 DIsoHcdih 40403 ocHcoch 40522

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-riotaBAD 38127

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-tpos 8214 df-undef 8261 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-0g 17392 df-proset 18253 df-poset 18271 df-plt 18288 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-p0 18383 df-p1 18384 df-lat 18390 df-clat 18457 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19040 df-cntz 19223 df-lsm 19546 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-drng 20503 df-lmod 20617 df-lss 20688 df-lsp 20728 df-lvec 20859 df-lsatoms 38150 df-oposet 38350 df-ol 38352 df-oml 38353 df-covers 38440 df-ats 38441 df-atl 38472 df-cvlat 38496 df-hlat 38525 df-llines 38673 df-lplanes 38674 df-lvols 38675 df-lines 38676 df-psubsp 38678 df-pmap 38679 df-padd 38971 df-lhyp 39163 df-laut 39164 df-ldil 39279 df-ltrn 39280 df-trl 39334 df-tendo 39930 df-edring 39932 df-disoa 40204 df-dvech 40254 df-dib 40314 df-dic 40348 df-dih 40404 df-doch 40523

This theorem is referenced by: dochnel2 40567 djhexmid 40586 dochexmidlem1 40635 lcfrlem23 40740

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