dochnoncon - Mathbox for Norm Megill

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Theorem dochnoncon 41393

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 2737	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
2		dochnoncon.s	. . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑈)
3	1, 2	lssss 20934	. . . . 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 41368	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋)))
8	3, 7	sylan2 593	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋)))
9	8	ssrind 4244	. . 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 2737	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2737	. . . . . . . . . 10 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
13		eqid 2737	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
14	11, 4, 12, 5, 13	dihf11 41269	. . . . . . . . 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 6859	. . . . . . . 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 41355	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
19	3, 18	sylan2 593	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊))
20	4, 5, 12, 13	dihrnlss 41279	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈))
21	19, 20	syldan 591	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ( ⊥ ‘𝑋) ∈ (LSubSp‘𝑈))
22	1, 13	lssss 20934	. . . . . . . . 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 41355	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
25	23, 24	syldan 591	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
26		f1ocnvdm 7305	. . . . . . 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 584	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))) ∈ (Base‘𝐾))
28		hlop 39363	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
29	28	ad2antrr 726	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → 𝐾 ∈ OP)
30		eqid 2737	. . . . . . . 8 ⊢ (oc‘𝐾) = (oc‘𝐾)
31	11, 30	opoccl 39195	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ (^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))) ∈ (Base‘𝐾))
32	29, 27, 31	syl2anc 584	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))) ∈ (Base‘𝐾))
33		eqid 2737	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
34	11, 33, 4, 12	dihmeet 41345	. . . . . 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 1373	. . . . 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 2737	. . . . . . . 8 ⊢ (0.‘𝐾) = (0.‘𝐾)
37	11, 30, 33, 36	opnoncon 39209	. . . . . . 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 584	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ((^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))(meet‘𝐾)((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))))) = (0.‘𝐾))
39	38	fveq2d 6910	. . . . 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 2779	. . . 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 41275	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (((DIsoH‘𝐾)‘𝑊)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))) = ( ⊥ ‘( ⊥ ‘𝑋)))
42	25, 41	syldan 591	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (((DIsoH‘𝐾)‘𝑊)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋)))) = ( ⊥ ‘( ⊥ ‘𝑋)))
43	30, 4, 12, 6	dochvalr 41359	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))))))
44	25, 43	syldan 591	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))))))
45	4, 12, 6	dochoc 41369	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑋))
46	19, 45	syldan 591	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑋))) = ( ⊥ ‘𝑋))
47	44, 46	eqtr3d 2779	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))))) = ( ⊥ ‘𝑋))
48	42, 47	ineq12d 4221	. . . 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 41282	. . . . 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 2785	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (( ⊥ ‘( ⊥ ‘𝑋)) ∩ ( ⊥ ‘𝑋)) = { 0 })
53	9, 52	sseqtrd 4020	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) ⊆ { 0 })
54	4, 5, 10	dvhlmod 41112	. . 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 41279	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘𝑋) ∈ 𝑆)
57	19, 56	syldan 591	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ( ⊥ ‘𝑋) ∈ 𝑆)
58	2	lssincl 20963	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ ( ⊥ ‘𝑋) ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) ∈ 𝑆)
59	54, 55, 57, 58	syl3anc 1373	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) ∈ 𝑆)
60	49, 2	lss0ss 20947	. . 3 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 ∩ ( ⊥ ‘𝑋)) ∈ 𝑆) → { 0 } ⊆ (𝑋 ∩ ( ⊥ ‘𝑋)))
61	54, 59, 60	syl2anc 584	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → { 0 } ⊆ (𝑋 ∩ ( ⊥ ‘𝑋)))
62	53, 61	eqssd 4001	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 2108 ∩ cin 3950 ⊆ wss 3951 {csn 4626 ^◡ccnv 5684 ran crn 5686 –1-1→wf1 6558 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 occoc 17305 0_gc0g 17484 meetcmee 18358 0.cp0 18468 LModclmod 20858 LSubSpclss 20929 OPcops 39173 HLchlt 39351 LHypclh 39986 DVecHcdvh 41080 DIsoHcdih 41230 ocHcoch 41349

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-riotaBAD 38954

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-undef 8298 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17486 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lvec 21102 df-lsatoms 38977 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161 df-tendo 40757 df-edring 40759 df-disoa 41031 df-dvech 41081 df-dib 41141 df-dic 41175 df-dih 41231 df-doch 41350

This theorem is referenced by: dochnel2 41394 djhexmid 41413 dochexmidlem1 41462 lcfrlem23 41567

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