dochnoncon - Mathbox for Norm Megill

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Theorem dochnoncon 39332

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 2738	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
2		dochnoncon.s	. . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑈)
3	1, 2	lssss 20113	. . . . 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 39307	. . . . 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 4166	. . 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 2738	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2738	. . . . . . . . . 10 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
13		eqid 2738	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
14	11, 4, 12, 5, 13	dihf11 39208	. . . . . . . . 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 6711	. . . . . . . 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 39294	. . . . . . . . . . 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 39218	. . . . . . . . . 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 20113	. . . . . . . . 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 39294	. . . . . . . 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 7137	. . . . . . 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 37303	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
29	28	ad2antrr 722	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → 𝐾 ∈ OP)
30		eqid 2738	. . . . . . . 8 ⊢ (oc‘𝐾) = (oc‘𝐾)
31	11, 30	opoccl 37135	. . . . . . 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 2738	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
34	11, 33, 4, 12	dihmeet 39284	. . . . . 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 1369	. . . . 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 2738	. . . . . . . 8 ⊢ (0.‘𝐾) = (0.‘𝐾)
37	11, 30, 33, 36	opnoncon 37149	. . . . . . 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 6760	. . . . 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 2780	. . . 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 39214	. . . . . 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 39298	. . . . . . 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 39308	. . . . . . 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 2780	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))))) = ( ⊥ ‘𝑋))
48	42, 47	ineq12d 4144	. . . 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 39221	. . . . 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 2786	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (( ⊥ ‘( ⊥ ‘𝑋)) ∩ ( ⊥ ‘𝑋)) = { 0 })
53	9, 52	sseqtrd 3957	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) ⊆ { 0 })
54	4, 5, 10	dvhlmod 39051	. . 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 39218	. . . . 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 20142	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ ( ⊥ ‘𝑋) ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) ∈ 𝑆)
59	54, 55, 57, 58	syl3anc 1369	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) ∈ 𝑆)
60	49, 2	lss0ss 20125	. . 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 3934	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) = { 0 })

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∩ cin 3882 ⊆ wss 3883 {csn 4558 ^◡ccnv 5579 ran crn 5581 –1-1→wf1 6415 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 occoc 16896 0_gc0g 17067 meetcmee 17945 0.cp0 18056 LModclmod 20038 LSubSpclss 20108 OPcops 37113 HLchlt 37291 LHypclh 37925 DVecHcdvh 39019 DIsoHcdih 39169 ocHcoch 39288

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-riotaBAD 36894

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-undef 8060 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-0g 17069 df-proset 17928 df-poset 17946 df-plt 17963 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-p0 18058 df-p1 18059 df-lat 18065 df-clat 18132 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-subg 18667 df-cntz 18838 df-lsm 19156 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-drng 19908 df-lmod 20040 df-lss 20109 df-lsp 20149 df-lvec 20280 df-lsatoms 36917 df-oposet 37117 df-ol 37119 df-oml 37120 df-covers 37207 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 df-llines 37439 df-lplanes 37440 df-lvols 37441 df-lines 37442 df-psubsp 37444 df-pmap 37445 df-padd 37737 df-lhyp 37929 df-laut 37930 df-ldil 38045 df-ltrn 38046 df-trl 38100 df-tendo 38696 df-edring 38698 df-disoa 38970 df-dvech 39020 df-dib 39080 df-dic 39114 df-dih 39170 df-doch 39289

This theorem is referenced by: dochnel2 39333 djhexmid 39352 dochexmidlem1 39401 lcfrlem23 39506

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