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Theorem opphllem2 29084

Description: Lemma for opphl 29090. 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 486	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐷 ∈ ran 𝐿)
8		opphl.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
9	8	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐺 ∈ TarskiG)
10		opphllem1.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
11	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐶 ∈ 𝑃)
12		opphllem1.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
13	12	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐵 ∈ 𝑃)
14		opphllem1.s	. . . 4 ⊢ 𝑆 = ((pInvG‘𝐺)‘𝑀)
15		eqid 2765	. . . . 5 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
16		opphllem1.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ 𝐷)
17	1, 5, 3, 8, 6, 16	tglnpt 28873	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑃)
18	17	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝑀 ∈ 𝑃)
19	1, 2, 3, 5, 15, 9, 18, 14, 13	mircl 28993	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐵) ∈ 𝑃)
20	16	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝑀 ∈ 𝐷)
21		opphllem1.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝐷)
22	21	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝑅 ∈ 𝐷)
23	1, 2, 3, 5, 15, 9, 14, 7, 20, 22	mirln 29008	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝑅) ∈ 𝐷)
24		simpr 490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
25		simplr 781	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝐷)
26	24, 25	eqeltrd 2865	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐷)
27	8	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐺 ∈ TarskiG)
28	12	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑃)
29	1, 5, 3, 8, 6, 21	tglnpt 28873	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
30	29	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝑅 ∈ 𝑃)
31		opphllem1.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
32	31	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝑃)
33		opphllem1.y	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ≠ 𝑅)
34	33	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐵 ≠ 𝑅)
35	34	necomd 3015	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝑅 ≠ 𝐵)
36		simpllr 788	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ (𝑅𝐼𝐵))
37	1, 3, 5, 27, 30, 28, 32, 35, 36	btwnlng1 28947	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ (𝑅𝐿𝐵))
38	1, 3, 5, 27, 28, 30, 32, 34, 37	lncom 28950	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ (𝐵𝐿𝑅))
39	6	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐷 ∈ ran 𝐿)
40		simplr 781	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝐷)
41	21	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝑅 ∈ 𝐷)
42	1, 3, 5, 27, 28, 30, 34, 34, 39, 40, 41	tglinethru 28964	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐷 = (𝐵𝐿𝑅))
43	38, 42	eleqtrrd 2868	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) ∧ 𝐴 ≠ 𝐵) → 𝐴 ∈ 𝐷)
44	26, 43	pm2.61dane 3047	. . . . . . 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 29077	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)
47	46	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ 𝐵 ∈ 𝐷) → ¬ 𝐴 ∈ 𝐷)
48	44, 47	pm2.65da 829	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → ¬ 𝐵 ∈ 𝐷)
49	9	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝐺 ∈ TarskiG)
50	18	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝑀 ∈ 𝑃)
51	13	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝐵 ∈ 𝑃)
52	1, 2, 3, 5, 15, 49, 50, 14, 51	mirmir 28994	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → (𝑆‘(𝑆‘𝐵)) = 𝐵)
53	7	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝐷 ∈ ran 𝐿)
54	20	adantr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝑀 ∈ 𝐷)
55		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → (𝑆‘𝐵) ∈ 𝐷)
56	1, 2, 3, 5, 15, 49, 14, 53, 54, 55	mirln 29008	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → (𝑆‘(𝑆‘𝐵)) ∈ 𝐷)
57	52, 56	eqeltrrd 2866	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) ∧ (𝑆‘𝐵) ∈ 𝐷) → 𝐵 ∈ 𝐷)
58	48, 57	mtand 828	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → ¬ (𝑆‘𝐵) ∈ 𝐷)
59	1, 2, 3, 5, 15, 9, 18, 14, 13	mirbtwn 28990	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝑀 ∈ ((𝑆‘𝐵)𝐼𝐵))
60	1, 2, 3, 4, 19, 13, 20, 58, 48, 59	islnoppd 29076	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐵)𝑂𝐵)
61		eqidd 2766	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐵) = (𝑆‘𝐵))
62		nelne2 3058	. . . . . 6 ⊢ (((𝑆‘𝑅) ∈ 𝐷 ∧ ¬ (𝑆‘𝐵) ∈ 𝐷) → (𝑆‘𝑅) ≠ (𝑆‘𝐵))
63	23, 58, 62	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝑅) ≠ (𝑆‘𝐵))
64	63	necomd 3015	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐵) ≠ (𝑆‘𝑅))
65	1, 2, 3, 4, 5, 6, 8, 31, 10, 45	oppne2 29078	. . . . . . 7 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐷)
66	65	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → ¬ 𝐶 ∈ 𝐷)
67		nelne2 3058	. . . . . 6 ⊢ (((𝑆‘𝑅) ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷) → (𝑆‘𝑅) ≠ 𝐶)
68	23, 66, 67	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝑅) ≠ 𝐶)
69	68	necomd 3015	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐶 ≠ (𝑆‘𝑅))
70		opphllem1.n	. . . . . . . 8 ⊢ (𝜑 → 𝐴 = (𝑆‘𝐶))
71	70	eqcomd 2771	. . . . . . 7 ⊢ (𝜑 → (𝑆‘𝐶) = 𝐴)
72	1, 2, 3, 5, 15, 8, 17, 14, 10, 71	mircom 28995	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝐴) = 𝐶)
73	72	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐴) = 𝐶)
74	29	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝑅 ∈ 𝑃)
75	31	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐴 ∈ 𝑃)
76		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐴 ∈ (𝑅𝐼𝐵))
77	1, 2, 3, 5, 15, 9, 18, 14, 74, 75, 13, 76	mirbtwni 29003	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → (𝑆‘𝐴) ∈ ((𝑆‘𝑅)𝐼(𝑆‘𝐵)))
78	73, 77	eqeltrrd 2866	. . . 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 29083	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐶𝑂𝐵)
80	1, 2, 3, 4, 5, 7, 9, 11, 13, 79	oppcom 29080	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ (𝑅𝐼𝐵)) → 𝐵𝑂𝐶)
81	6	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐷 ∈ ran 𝐿)
82	8	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐺 ∈ TarskiG)
83	31	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐴 ∈ 𝑃)
84	12	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐵 ∈ 𝑃)
85	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐶 ∈ 𝑃)
86	21	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝑅 ∈ 𝐷)
87	45	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐴𝑂𝐶)
88	16	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝑀 ∈ 𝐷)
89	70	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐴 = (𝑆‘𝐶))
90		opphllem1.x	. . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝑅)
91	90	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐴 ≠ 𝑅)
92	33	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐵 ≠ 𝑅)
93		simpr 490	. . 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 29083	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ (𝑅𝐼𝐴)) → 𝐵𝑂𝐶)
95		opphllem2.z	. 2 ⊢ (𝜑 → (𝐴 ∈ (𝑅𝐼𝐵) ∨ 𝐵 ∈ (𝑅𝐼𝐴)))
96	80, 94, 95	mpjaodan 973	1 ⊢ (𝜑 → 𝐵𝑂𝐶)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∃wrex 3091 ∖ cdif 3903 class class class wbr 5111 {copab 5175 ran crn 5664 ‘cfv 6540 (class class class)co 7419 Basecbs 17295 distcds 17345 TarskiGcstrkg 28751 Itvcitv 28757 LineGclng 28758 pInvGcmir 28984

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11175 ax-resscn 11176 ax-1cn 11177 ax-icn 11178 ax-addcl 11179 ax-addrcl 11180 ax-mulcl 11181 ax-mulrcl 11182 ax-mulcom 11183 ax-addass 11184 ax-mulass 11185 ax-distr 11186 ax-i2m1 11187 ax-1ne0 11188 ax-1rid 11189 ax-rnegex 11190 ax-rrecex 11191 ax-cnre 11192 ax-pre-lttri 11193 ax-pre-lttrn 11194 ax-pre-ltadd 11195 ax-pre-mulgt0 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-oadd 8463 df-er 8700 df-pm 8833 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-dju 9903 df-card 9941 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11462 df-neg 11463 df-nn 12253 df-2 12322 df-3 12323 df-n0 12524 df-xnn0 12597 df-z 12611 df-uz 12883 df-fz 13556 df-fzo 13704 df-hash 14389 df-word 14573 df-concat 14630 df-s1 14657 df-s2 14913 df-s3 14914 df-trkgc 28772 df-trkgb 28773 df-trkgcb 28774 df-trkg 28777 df-cgrg 28835 df-mir 28985

This theorem is used by: opphllem4 29086 opphl 29090

W3C validator