dochnoncon - Mathbox for Norm Megill

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Theorem dochnoncon 40354

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 2732	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
2		dochnoncon.s	. . . . . 6 ⊢ 𝑆 = (LSubSp‘𝑈)
3	1, 2	lssss 20552	. . . . 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 40329	. . . . 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 4235	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) ⊆ (( ⊥ ‘( ⊥ ‘𝑋)) ∩ ( ⊥ ‘𝑋)))
10		simpl 483	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		eqid 2732	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
12		eqid 2732	. . . . . . . . . 10 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
13		eqid 2732	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
14	11, 4, 12, 5, 13	dihf11 40230	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((DIsoH‘𝐾)‘𝑊):(Base‘𝐾)–1-1→(LSubSp‘𝑈))
15	14	adantr 481	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → ((DIsoH‘𝐾)‘𝑊):(Base‘𝐾)–1-1→(LSubSp‘𝑈))
16		f1f1orn 6844	. . . . . . . 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 40316	. . . . . . . . . . 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 40240	. . . . . . . . . 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 20552	. . . . . . . . 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 40316	. . . . . . . 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 7285	. . . . . . 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 38324	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
29	28	ad2antrr 724	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → 𝐾 ∈ OP)
30		eqid 2732	. . . . . . . 8 ⊢ (oc‘𝐾) = (oc‘𝐾)
31	11, 30	opoccl 38156	. . . . . . 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 2732	. . . . . . 7 ⊢ (meet‘𝐾) = (meet‘𝐾)
34	11, 33, 4, 12	dihmeet 40306	. . . . . 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 2732	. . . . . . . 8 ⊢ (0.‘𝐾) = (0.‘𝐾)
37	11, 30, 33, 36	opnoncon 38170	. . . . . . 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 6895	. . . . 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 2774	. . . 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 40236	. . . . . 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 40320	. . . . . . 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 40330	. . . . . . 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 2774	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (((DIsoH‘𝐾)‘𝑊)‘((oc‘𝐾)‘(^◡((DIsoH‘𝐾)‘𝑊)‘( ⊥ ‘( ⊥ ‘𝑋))))) = ( ⊥ ‘𝑋))
48	42, 47	ineq12d 4213	. . . 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 40243	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((DIsoH‘𝐾)‘𝑊)‘(0.‘𝐾)) = { 0 })
51	50	adantr 481	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (((DIsoH‘𝐾)‘𝑊)‘(0.‘𝐾)) = { 0 })
52	40, 48, 51	3eqtr3d 2780	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (( ⊥ ‘( ⊥ ‘𝑋)) ∩ ( ⊥ ‘𝑋)) = { 0 })
53	9, 52	sseqtrd 4022	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) ⊆ { 0 })
54	4, 5, 10	dvhlmod 40073	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → 𝑈 ∈ LMod)
55		simpr 485	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
56	4, 5, 12, 2	dihrnlss 40240	. . . . 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 20581	. . . 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 20564	. . 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 3999	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) = { 0 })

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∩ cin 3947 ⊆ wss 3948 {csn 4628 ^◡ccnv 5675 ran crn 5677 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7411 Basecbs 17146 occoc 17207 0_gc0g 17387 meetcmee 18267 0.cp0 18378 LModclmod 20475 LSubSpclss 20547 OPcops 38134 HLchlt 38312 LHypclh 38947 DVecHcdvh 40041 DIsoHcdih 40191 ocHcoch 40310

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-riotaBAD 37915

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-undef 8260 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-n0 12475 df-z 12561 df-uz 12825 df-fz 13487 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-sca 17215 df-vsca 17216 df-0g 17389 df-proset 18250 df-poset 18268 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18387 df-clat 18454 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-grp 18824 df-minusg 18825 df-sbg 18826 df-subg 19005 df-cntz 19183 df-lsm 19506 df-cmn 19652 df-abl 19653 df-mgp 19990 df-ur 20007 df-ring 20060 df-oppr 20154 df-dvdsr 20175 df-unit 20176 df-invr 20206 df-dvr 20219 df-drng 20363 df-lmod 20477 df-lss 20548 df-lsp 20588 df-lvec 20719 df-lsatoms 37938 df-oposet 38138 df-ol 38140 df-oml 38141 df-covers 38228 df-ats 38229 df-atl 38260 df-cvlat 38284 df-hlat 38313 df-llines 38461 df-lplanes 38462 df-lvols 38463 df-lines 38464 df-psubsp 38466 df-pmap 38467 df-padd 38759 df-lhyp 38951 df-laut 38952 df-ldil 39067 df-ltrn 39068 df-trl 39122 df-tendo 39718 df-edring 39720 df-disoa 39992 df-dvech 40042 df-dib 40102 df-dic 40136 df-dih 40192 df-doch 40311

This theorem is referenced by: dochnel2 40355 djhexmid 40374 dochexmidlem1 40423 lcfrlem23 40528

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