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Theorem opphllem2 28835

Description: Lemma for opphl 28841. 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 2739	. . . . 5 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
16		opphllem1.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ 𝐷)
17	1, 5, 3, 8, 6, 16	tglnpt 28636	. . . . . 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 28748	. . . 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 28763	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝑅) ∈ 𝐷)
24		simpr 485	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
25		simplr 774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝐷)
26	24, 25	eqeltrd 2839	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐷)
27	8	ad3antrrr 736	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐺 ∈ TarskiG)
28	12	ad3antrrr 736	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑃)
29	1, 5, 3, 8, 6, 21	tglnpt 28636	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
30	29	ad3antrrr 736	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝑅 ∈ 𝑃)
31		opphllem1.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
32	31	ad3antrrr 736	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑃)
33		opphllem1.y	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ≠ 𝑅)
34	33	ad3antrrr 736	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐵 ≠ 𝑅)
35	34	necomd 2989	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝑅 ≠ 𝐵)
36		simpllr 781	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ (𝑅𝐼𝐵))
37	1, 3, 5, 27, 30, 28, 32, 35, 36	btwnlng1 28706	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ (𝑅𝐿𝐵))
38	1, 3, 5, 27, 28, 30, 32, 34, 37	lncom 28709	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ (𝐵𝐿𝑅))
39	6	ad3antrrr 736	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐷 ∈ ran 𝐿)
40		simplr 774	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝐷)
41	21	ad3antrrr 736	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝑅 ∈ 𝐷)
42	1, 3, 5, 27, 28, 30, 34, 34, 39, 40, 41	tglinethru 28723	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐷 = (𝐵𝐿𝑅))
43	38, 42	eleqtrrd 2842	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝐷)
44	26, 43	pm2.61dane 3021	. . . . . . 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 28828	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
47	46	ad2antrr 732	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) → ¬ 𝐴 ∈ 𝐷)
48	44, 47	pm2.65da 822	. . . . . 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 28749	. . . . . . 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 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → (𝑆‘𝐵) ∈ 𝐷)
56	1, 2, 3, 5, 15, 49, 14, 53, 54, 55	mirln 28763	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → (𝑆‘(𝑆‘𝐵)) ∈ 𝐷)
57	52, 56	eqeltrrd 2840	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝐵 ∈ 𝐷)
58	48, 57	mtand 821	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → ¬ (𝑆‘𝐵) ∈ 𝐷)
59	1, 2, 3, 5, 15, 9, 18, 14, 13	mirbtwn 28745	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝑀 ∈ ((𝑆‘𝐵)𝐼𝐵))
60	1, 2, 3, 4, 19, 13, 20, 58, 48, 59	islnoppd 28827	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐵)𝑂𝐵)
61		eqidd 2740	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐵) = (𝑆‘𝐵))
62		nelne2 3032	. . . . . 6 ⊢ (((𝑆‘𝑅) ∈ 𝐷 ∧ ¬ (𝑆‘𝐵) ∈ 𝐷) → (𝑆‘𝑅) ≠ (𝑆‘𝐵))
63	23, 58, 62	syl2anc 590	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝑅) ≠ (𝑆‘𝐵))
64	63	necomd 2989	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐵) ≠ (𝑆‘𝑅))
65	1, 2, 3, 4, 5, 6, 8, 31, 10, 45	oppne2 28829	. . . . . . 7 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐷)
66	65	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → ¬ 𝐶 ∈ 𝐷)
67		nelne2 3032	. . . . . 6 ⊢ (((𝑆‘𝑅) ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) → (𝑆‘𝑅) ≠ 𝐶)
68	23, 66, 67	syl2anc 590	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝑅) ≠ 𝐶)
69	68	necomd 2989	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐶 ≠ (𝑆‘𝑅))
70		opphllem1.n	. . . . . . . 8 ⊢ (𝜑 → 𝐴 = (𝑆‘𝐶))
71	70	eqcomd 2745	. . . . . . 7 ⊢ (𝜑 → (𝑆‘𝐶) = 𝐴)
72	1, 2, 3, 5, 15, 8, 17, 14, 10, 71	mircom 28750	. . . . . 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 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐴 ∈ (𝑅𝐼𝐵))
77	1, 2, 3, 5, 15, 9, 18, 14, 74, 75, 13, 76	mirbtwni 28758	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐴) ∈ ((𝑆‘𝑅)𝐼(𝑆‘𝐵)))
78	73, 77	eqeltrrd 2840	. . . 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 28834	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐶𝑂𝐵)
80	1, 2, 3, 4, 5, 7, 9, 11, 13, 79	oppcom 28831	. 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 485	. . 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 28834	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐵𝑂𝐶)
95		opphllem2.z	. 2 ⊢ (𝜑 → (𝐴 ∈ (𝑅𝐼𝐵) ∨ 𝐵 ∈ (𝑅𝐼𝐴)))
96	80, 94, 95	mpjaodan 966	1 ⊢ (𝜑 → 𝐵𝑂𝐶)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 853 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 ∃wrex 3063 ∖ cdif 3880 class class class wbr 5073 {copab 5135 ran crn 5620 ‘cfv 6486 (class class class)co 7357 Basecbs 17171 distcds 17221 TarskiGcstrkg 28514 Itvcitv 28520 LineGclng 28521 pInvGcmir 28739

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7808 df-1st 7932 df-2nd 7933 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-oadd 8400 df-er 8634 df-pm 8767 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-dju 9817 df-card 9855 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-nn 12167 df-2 12236 df-3 12237 df-n0 12430 df-xnn0 12503 df-z 12517 df-uz 12781 df-fz 13454 df-fzo 13601 df-hash 14285 df-word 14468 df-concat 14525 df-s1 14551 df-s2 14802 df-s3 14803 df-trkgc 28535 df-trkgb 28536 df-trkgcb 28537 df-trkg 28540 df-cgrg 28598 df-mir 28740

This theorem is referenced by: opphllem4 28837 opphl 28841

W3C validator