opphllem2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > opphllem2		Structured version Visualization version GIF version

Theorem opphllem2 25640

Description: Lemma for opphl 25646. Lemma 9.3 of [Schwabhauser] p. 68. (Contributed by Thierry Arnoux, 21-Dec-2019.)

**Hypotheses**
Ref	Expression
hpg.p	⊢ 𝑃 = (Base‘𝐺)
hpg.d	⊢ − = (dist‘𝐺)
hpg.i	⊢ 𝐼 = (Itv‘𝐺)
hpg.o	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
opphl.l	⊢ 𝐿 = (LineG‘𝐺)
opphl.d	⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
opphl.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
opphllem1.s	⊢ 𝑆 = ((pInvG‘𝐺)‘𝑀)
opphllem1.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
opphllem1.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
opphllem1.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
opphllem1.r	⊢ (𝜑 → 𝑅 ∈ 𝐷)
opphllem1.o	⊢ (𝜑 → 𝐴𝑂𝐶)
opphllem1.m	⊢ (𝜑 → 𝑀 ∈ 𝐷)
opphllem1.n	⊢ (𝜑 → 𝐴 = (𝑆‘𝐶))
opphllem1.x	⊢ (𝜑 → 𝐴 ≠ 𝑅)
opphllem1.y	⊢ (𝜑 → 𝐵 ≠ 𝑅)
opphllem2.z	⊢ (𝜑 → (𝐴 ∈ (𝑅𝐼𝐵) ∨ 𝐵 ∈ (𝑅𝐼𝐴)))

**Assertion**
Ref	Expression
opphllem2	⊢ (𝜑 → 𝐵𝑂𝐶)

Distinct variable groups: 𝐷,𝑎,𝑏 𝐼,𝑎,𝑏 𝑃,𝑎,𝑏 𝑡,𝐴 𝑡,𝐵 𝑡,𝐷 𝑡,𝑅 𝑡,𝐶 𝑡,𝐺 𝑡,𝐿 𝑡,𝐼 𝑡,𝑀 𝑡,𝑂 𝑡,𝑃 𝑡,𝑆 𝜑,𝑡 𝑡, − 𝑡,𝑎,𝑏 Allowed substitution hints: 𝜑(𝑎,𝑏) 𝐴(𝑎,𝑏) 𝐵(𝑎,𝑏) 𝐶(𝑎,𝑏) 𝑅(𝑎,𝑏) 𝑆(𝑎,𝑏) 𝐺(𝑎,𝑏) 𝐿(𝑎,𝑏) 𝑀(𝑎,𝑏) − (𝑎,𝑏) 𝑂(𝑎,𝑏)

**Proof of Theorem opphllem2**
Step	Hyp	Ref	Expression
1		hpg.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		hpg.d	. . 3 ⊢ − = (dist‘𝐺)
3		hpg.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		hpg.o	. . 3 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
5		opphl.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
6		opphl.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
7	6	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐷 ∈ ran 𝐿)
8		opphl.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
9	8	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐺 ∈ TarskiG)
10		opphllem1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11	10	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐶 ∈ 𝑃)
12		opphllem1.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
13	12	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐵 ∈ 𝑃)
14		opphllem1.s	. . . 4 ⊢ 𝑆 = ((pInvG‘𝐺)‘𝑀)
15		eqid 2622	. . . . 5 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
16		opphllem1.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ 𝐷)
17	1, 5, 3, 8, 6, 16	tglnpt 25444	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑃)
18	17	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝑀 ∈ 𝑃)
19	1, 2, 3, 5, 15, 9, 18, 14, 13	mircl 25556	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐵) ∈ 𝑃)
20	16	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝑀 ∈ 𝐷)
21		opphllem1.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝐷)
22	21	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝑅 ∈ 𝐷)
23	1, 2, 3, 5, 15, 9, 14, 7, 20, 22	mirln 25571	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝑅) ∈ 𝐷)
24		simpr 477	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
25		simplr 792	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝐷)
26	24, 25	eqeltrd 2701	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐷)
27	8	ad3antrrr 766	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐺 ∈ TarskiG)
28	12	ad3antrrr 766	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑃)
29	1, 5, 3, 8, 6, 21	tglnpt 25444	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
30	29	ad3antrrr 766	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝑅 ∈ 𝑃)
31		opphllem1.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
32	31	ad3antrrr 766	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑃)
33		opphllem1.y	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ≠ 𝑅)
34	33	ad3antrrr 766	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐵 ≠ 𝑅)
35	34	necomd 2849	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝑅 ≠ 𝐵)
36		simpllr 799	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ (𝑅𝐼𝐵))
37	1, 3, 5, 27, 30, 28, 32, 35, 36	btwnlng1 25514	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ (𝑅𝐿𝐵))
38	1, 3, 5, 27, 28, 30, 32, 34, 37	lncom 25517	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ (𝐵𝐿𝑅))
39	6	ad3antrrr 766	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐷 ∈ ran 𝐿)
40		simplr 792	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝐷)
41	21	ad3antrrr 766	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝑅 ∈ 𝐷)
42	1, 3, 5, 27, 28, 30, 34, 34, 39, 40, 41	tglinethru 25531	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐷 = (𝐵𝐿𝑅))
43	38, 42	eleqtrrd 2704	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝐷)
44	26, 43	pm2.61dane 2881	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) → 𝐴 ∈ 𝐷)
45		opphllem1.o	. . . . . . . . 9 ⊢ (𝜑 → 𝐴𝑂𝐶)
46	1, 2, 3, 4, 5, 6, 8, 31, 10, 45	oppne1 25633	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
47	46	ad2antrr 762	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) → ¬ 𝐴 ∈ 𝐷)
48	44, 47	pm2.65da 600	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → ¬ 𝐵 ∈ 𝐷)
49	9	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝐺 ∈ TarskiG)
50	18	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝑀 ∈ 𝑃)
51	13	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝐵 ∈ 𝑃)
52	1, 2, 3, 5, 15, 49, 50, 14, 51	mirmir 25557	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → (𝑆‘(𝑆‘𝐵)) = 𝐵)
53	7	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
54	20	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝑀 ∈ 𝐷)
55		simpr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → (𝑆‘𝐵) ∈ 𝐷)
56	1, 2, 3, 5, 15, 49, 14, 53, 54, 55	mirln 25571	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → (𝑆‘(𝑆‘𝐵)) ∈ 𝐷)
57	52, 56	eqeltrrd 2702	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝐵 ∈ 𝐷)
58	48, 57	mtand 691	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → ¬ (𝑆‘𝐵) ∈ 𝐷)
59	1, 2, 3, 5, 15, 9, 18, 14, 13	mirbtwn 25553	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝑀 ∈ ((𝑆‘𝐵)𝐼𝐵))
60	1, 2, 3, 4, 19, 13, 20, 58, 48, 59	islnoppd 25632	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐵)𝑂𝐵)
61		eqidd 2623	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐵) = (𝑆‘𝐵))
62		nelne2 2891	. . . . . 6 ⊢ (((𝑆‘𝑅) ∈ 𝐷 ∧ ¬ (𝑆‘𝐵) ∈ 𝐷) → (𝑆‘𝑅) ≠ (𝑆‘𝐵))
63	23, 58, 62	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝑅) ≠ (𝑆‘𝐵))
64	63	necomd 2849	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐵) ≠ (𝑆‘𝑅))
65	1, 2, 3, 4, 5, 6, 8, 31, 10, 45	oppne2 25634	. . . . . . 7 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐷)
66	65	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → ¬ 𝐶 ∈ 𝐷)
67		nelne2 2891	. . . . . 6 ⊢ (((𝑆‘𝑅) ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) → (𝑆‘𝑅) ≠ 𝐶)
68	23, 66, 67	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝑅) ≠ 𝐶)
69	68	necomd 2849	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐶 ≠ (𝑆‘𝑅))
70		opphllem1.n	. . . . . . . 8 ⊢ (𝜑 → 𝐴 = (𝑆‘𝐶))
71	70	eqcomd 2628	. . . . . . 7 ⊢ (𝜑 → (𝑆‘𝐶) = 𝐴)
72	1, 2, 3, 5, 15, 8, 17, 14, 10, 71	mircom 25558	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝐴) = 𝐶)
73	72	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐴) = 𝐶)
74	29	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝑅 ∈ 𝑃)
75	31	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐴 ∈ 𝑃)
76		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐴 ∈ (𝑅𝐼𝐵))
77	1, 2, 3, 5, 15, 9, 18, 14, 74, 75, 13, 76	mirbtwni 25566	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐴) ∈ ((𝑆‘𝑅)𝐼(𝑆‘𝐵)))
78	73, 77	eqeltrrd 2702	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐶 ∈ ((𝑆‘𝑅)𝐼(𝑆‘𝐵)))
79	1, 2, 3, 4, 5, 7, 9, 14, 19, 11, 13, 23, 60, 20, 61, 64, 69, 78	opphllem1 25639	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐶𝑂𝐵)
80	1, 2, 3, 4, 5, 7, 9, 11, 13, 79	oppcom 25636	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐵𝑂𝐶)
81	6	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐷 ∈ ran 𝐿)
82	8	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐺 ∈ TarskiG)
83	31	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐴 ∈ 𝑃)
84	12	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐵 ∈ 𝑃)
85	10	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐶 ∈ 𝑃)
86	21	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝑅 ∈ 𝐷)
87	45	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐴𝑂𝐶)
88	16	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝑀 ∈ 𝐷)
89	70	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐴 = (𝑆‘𝐶))
90		opphllem1.x	. . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝑅)
91	90	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐴 ≠ 𝑅)
92	33	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐵 ≠ 𝑅)
93		simpr 477	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐵 ∈ (𝑅𝐼𝐴))
94	1, 2, 3, 4, 5, 81, 82, 14, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93	opphllem1 25639	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐵𝑂𝐶)
95		opphllem2.z	. 2 ⊢ (𝜑 → (𝐴 ∈ (𝑅𝐼𝐵) ∨ 𝐵 ∈ (𝑅𝐼𝐴)))
96	80, 94, 95	mpjaodan 827	1 ⊢ (𝜑 → 𝐵𝑂𝐶)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 ∖ cdif 3571 class class class wbr 4653 {copab 4712 ran crn 5115 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 distcds 15950 TarskiGcstrkg 25329 Itvcitv 25335 LineGclng 25336 pInvGcmir 25547

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 df-s3 13594 df-trkgc 25347 df-trkgb 25348 df-trkgcb 25349 df-trkg 25352 df-cgrg 25406 df-mir 25548

This theorem is referenced by: opphllem4 25642 opphl 25646

W3C validator