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Theorem legtrid 28097

Description: Trichotomy law for the less-than relationship. Proposition 5.10 of [Schwabhauser] p. 42. (Contributed by Thierry Arnoux, 27-Jun-2019.)

**Hypotheses**
Ref	Expression
legval.p	⊢ 𝑃 = (Base‘𝐺)
legval.d	⊢ − = (dist‘𝐺)
legval.i	⊢ 𝐼 = (Itv‘𝐺)
legval.l	⊢ ≤ = (≤G‘𝐺)
legval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
legid.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
legid.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
legtrd.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
legtrd.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)

**Assertion**
Ref	Expression
legtrid	⊢ (𝜑 → ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∨ (𝐶 − 𝐷) ≤ (𝐴 − 𝐵)))

Proof of Theorem legtrid Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		legval.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		legval.d	. . . . 5 ⊢ − = (dist‘𝐺)
3		legval.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
4		legval.l	. . . . 5 ⊢ ≤ = (≤G‘𝐺)
5		legval.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6	5	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑃) = 1) → 𝐺 ∈ TarskiG)
7		legid.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
8	7	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑃) = 1) → 𝐴 ∈ 𝑃)
9		legid.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
10	9	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑃) = 1) → 𝐵 ∈ 𝑃)
11	1, 2, 3, 4, 6, 8, 10	legid 28093	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝑃) = 1) → (𝐴 − 𝐵) ≤ (𝐴 − 𝐵))
12		legtrd.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
13	12	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑃) = 1) → 𝐶 ∈ 𝑃)
14		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑃) = 1) → (♯‘𝑃) = 1)
15		legtrd.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
16	15	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (♯‘𝑃) = 1) → 𝐷 ∈ 𝑃)
17	1, 2, 3, 6, 8, 10, 13, 14, 16	tgldim0cgr 28011	. . . 4 ⊢ ((𝜑 ∧ (♯‘𝑃) = 1) → (𝐴 − 𝐵) = (𝐶 − 𝐷))
18	11, 17	breqtrd 5174	. . 3 ⊢ ((𝜑 ∧ (♯‘𝑃) = 1) → (𝐴 − 𝐵) ≤ (𝐶 − 𝐷))
19	18	orcd 871	. 2 ⊢ ((𝜑 ∧ (♯‘𝑃) = 1) → ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∨ (𝐶 − 𝐷) ≤ (𝐴 − 𝐵)))
20	5	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) ∧ (𝑦 ∈ 𝑃 ∧ (𝐴 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)))) → 𝐺 ∈ TarskiG)
21		simplr 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) → 𝑥 ∈ 𝑃)
22	21	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) ∧ (𝑦 ∈ 𝑃 ∧ (𝐴 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)))) → 𝑥 ∈ 𝑃)
23	7	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) ∧ (𝑦 ∈ 𝑃 ∧ (𝐴 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)))) → 𝐴 ∈ 𝑃)
24	9	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) ∧ (𝑦 ∈ 𝑃 ∧ (𝐴 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)))) → 𝐵 ∈ 𝑃)
25		simprl 769	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) ∧ (𝑦 ∈ 𝑃 ∧ (𝐴 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)))) → 𝑦 ∈ 𝑃)
26		simplrr 776	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) ∧ (𝑦 ∈ 𝑃 ∧ (𝐴 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)))) → 𝐴 ≠ 𝑥)
27	26	necomd 2996	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) ∧ (𝑦 ∈ 𝑃 ∧ (𝐴 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)))) → 𝑥 ≠ 𝐴)
28		simplrl 775	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) ∧ (𝑦 ∈ 𝑃 ∧ (𝐴 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)))) → 𝐴 ∈ (𝐵𝐼𝑥))
29	1, 2, 3, 20, 24, 23, 22, 28	tgbtwncom 27994	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) ∧ (𝑦 ∈ 𝑃 ∧ (𝐴 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)))) → 𝐴 ∈ (𝑥𝐼𝐵))
30		simprrl 779	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) ∧ (𝑦 ∈ 𝑃 ∧ (𝐴 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)))) → 𝐴 ∈ (𝑥𝐼𝑦))
31	1, 3, 20, 22, 23, 24, 25, 27, 29, 30	tgbtwnconn2 28082	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) ∧ (𝑦 ∈ 𝑃 ∧ (𝐴 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)))) → (𝐵 ∈ (𝐴𝐼𝑦) ∨ 𝑦 ∈ (𝐴𝐼𝐵)))
32		simprrr 780	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) ∧ (𝑦 ∈ 𝑃 ∧ (𝐴 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)))) → (𝐴 − 𝑦) = (𝐶 − 𝐷))
33	31, 32	jca 512	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) ∧ (𝑦 ∈ 𝑃 ∧ (𝐴 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)))) → ((𝐵 ∈ (𝐴𝐼𝑦) ∨ 𝑦 ∈ (𝐴𝐼𝐵)) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)))
34	5	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) → 𝐺 ∈ TarskiG)
35	7	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) → 𝐴 ∈ 𝑃)
36	12	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) → 𝐶 ∈ 𝑃)
37	15	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) → 𝐷 ∈ 𝑃)
38	1, 2, 3, 34, 21, 35, 36, 37	axtgsegcon 27970	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) → ∃𝑦 ∈ 𝑃 (𝐴 ∈ (𝑥𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)))
39	33, 38	reximddv 3171	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) → ∃𝑦 ∈ 𝑃 ((𝐵 ∈ (𝐴𝐼𝑦) ∨ 𝑦 ∈ (𝐴𝐼𝐵)) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)))
40	39	adantllr 717	. . . . 5 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥)) → ∃𝑦 ∈ 𝑃 ((𝐵 ∈ (𝐴𝐼𝑦) ∨ 𝑦 ∈ (𝐴𝐼𝐵)) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)))
41	5	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 2 ≤ (♯‘𝑃)) → 𝐺 ∈ TarskiG)
42	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 2 ≤ (♯‘𝑃)) → 𝐵 ∈ 𝑃)
43	7	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 2 ≤ (♯‘𝑃)) → 𝐴 ∈ 𝑃)
44		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 2 ≤ (♯‘𝑃)) → 2 ≤ (♯‘𝑃))
45	1, 2, 3, 41, 42, 43, 44	tgbtwndiff 28012	. . . . 5 ⊢ ((𝜑 ∧ 2 ≤ (♯‘𝑃)) → ∃𝑥 ∈ 𝑃 (𝐴 ∈ (𝐵𝐼𝑥) ∧ 𝐴 ≠ 𝑥))
46	40, 45	r19.29a 3162	. . . 4 ⊢ ((𝜑 ∧ 2 ≤ (♯‘𝑃)) → ∃𝑦 ∈ 𝑃 ((𝐵 ∈ (𝐴𝐼𝑦) ∨ 𝑦 ∈ (𝐴𝐼𝐵)) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)))
47		andir 1007	. . . . . . 7 ⊢ (((𝐵 ∈ (𝐴𝐼𝑦) ∨ 𝑦 ∈ (𝐴𝐼𝐵)) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)) ↔ ((𝐵 ∈ (𝐴𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)) ∨ (𝑦 ∈ (𝐴𝐼𝐵) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷))))
48		eqcom 2739	. . . . . . . . 9 ⊢ ((𝐴 − 𝑦) = (𝐶 − 𝐷) ↔ (𝐶 − 𝐷) = (𝐴 − 𝑦))
49	48	anbi2i 623	. . . . . . . 8 ⊢ ((𝑦 ∈ (𝐴𝐼𝐵) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)) ↔ (𝑦 ∈ (𝐴𝐼𝐵) ∧ (𝐶 − 𝐷) = (𝐴 − 𝑦)))
50	49	orbi2i 911	. . . . . . 7 ⊢ (((𝐵 ∈ (𝐴𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)) ∨ (𝑦 ∈ (𝐴𝐼𝐵) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷))) ↔ ((𝐵 ∈ (𝐴𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)) ∨ (𝑦 ∈ (𝐴𝐼𝐵) ∧ (𝐶 − 𝐷) = (𝐴 − 𝑦))))
51	47, 50	bitri 274	. . . . . 6 ⊢ (((𝐵 ∈ (𝐴𝐼𝑦) ∨ 𝑦 ∈ (𝐴𝐼𝐵)) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)) ↔ ((𝐵 ∈ (𝐴𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)) ∨ (𝑦 ∈ (𝐴𝐼𝐵) ∧ (𝐶 − 𝐷) = (𝐴 − 𝑦))))
52	51	rexbii 3094	. . . . 5 ⊢ (∃𝑦 ∈ 𝑃 ((𝐵 ∈ (𝐴𝐼𝑦) ∨ 𝑦 ∈ (𝐴𝐼𝐵)) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)) ↔ ∃𝑦 ∈ 𝑃 ((𝐵 ∈ (𝐴𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)) ∨ (𝑦 ∈ (𝐴𝐼𝐵) ∧ (𝐶 − 𝐷) = (𝐴 − 𝑦))))
53		r19.43 3122	. . . . 5 ⊢ (∃𝑦 ∈ 𝑃 ((𝐵 ∈ (𝐴𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)) ∨ (𝑦 ∈ (𝐴𝐼𝐵) ∧ (𝐶 − 𝐷) = (𝐴 − 𝑦))) ↔ (∃𝑦 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)) ∨ ∃𝑦 ∈ 𝑃 (𝑦 ∈ (𝐴𝐼𝐵) ∧ (𝐶 − 𝐷) = (𝐴 − 𝑦))))
54	52, 53	bitri 274	. . . 4 ⊢ (∃𝑦 ∈ 𝑃 ((𝐵 ∈ (𝐴𝐼𝑦) ∨ 𝑦 ∈ (𝐴𝐼𝐵)) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)) ↔ (∃𝑦 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)) ∨ ∃𝑦 ∈ 𝑃 (𝑦 ∈ (𝐴𝐼𝐵) ∧ (𝐶 − 𝐷) = (𝐴 − 𝑦))))
55	46, 54	sylib 217	. . 3 ⊢ ((𝜑 ∧ 2 ≤ (♯‘𝑃)) → (∃𝑦 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)) ∨ ∃𝑦 ∈ 𝑃 (𝑦 ∈ (𝐴𝐼𝐵) ∧ (𝐶 − 𝐷) = (𝐴 − 𝑦))))
56	1, 2, 3, 4, 5, 7, 9, 12, 15	legov2 28092	. . . . 5 ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ↔ ∃𝑦 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷))))
57	1, 2, 3, 4, 5, 12, 15, 7, 9	legov 28091	. . . . 5 ⊢ (𝜑 → ((𝐶 − 𝐷) ≤ (𝐴 − 𝐵) ↔ ∃𝑦 ∈ 𝑃 (𝑦 ∈ (𝐴𝐼𝐵) ∧ (𝐶 − 𝐷) = (𝐴 − 𝑦))))
58	56, 57	orbi12d 917	. . . 4 ⊢ (𝜑 → (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∨ (𝐶 − 𝐷) ≤ (𝐴 − 𝐵)) ↔ (∃𝑦 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)) ∨ ∃𝑦 ∈ 𝑃 (𝑦 ∈ (𝐴𝐼𝐵) ∧ (𝐶 − 𝐷) = (𝐴 − 𝑦)))))
59	58	adantr 481	. . 3 ⊢ ((𝜑 ∧ 2 ≤ (♯‘𝑃)) → (((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∨ (𝐶 − 𝐷) ≤ (𝐴 − 𝐵)) ↔ (∃𝑦 ∈ 𝑃 (𝐵 ∈ (𝐴𝐼𝑦) ∧ (𝐴 − 𝑦) = (𝐶 − 𝐷)) ∨ ∃𝑦 ∈ 𝑃 (𝑦 ∈ (𝐴𝐼𝐵) ∧ (𝐶 − 𝐷) = (𝐴 − 𝑦)))))
60	55, 59	mpbird 256	. 2 ⊢ ((𝜑 ∧ 2 ≤ (♯‘𝑃)) → ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∨ (𝐶 − 𝐷) ≤ (𝐴 − 𝐵)))
61	1, 7	tgldimor 28008	. 2 ⊢ (𝜑 → ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃)))
62	19, 60, 61	mpjaodan 957	1 ⊢ (𝜑 → ((𝐴 − 𝐵) ≤ (𝐶 − 𝐷) ∨ (𝐶 − 𝐷) ≤ (𝐴 − 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 class class class wbr 5148 ‘cfv 6543 (class class class)co 7411 1c1 11113 ≤ cle 11253 2c2 12271 ♯chash 14294 Basecbs 17148 distcds 17210 TarskiGcstrkg 27933 Itvcitv 27939 ≤Gcleg 28088

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

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-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-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-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9898 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-xnn0 12549 df-z 12563 df-uz 12827 df-fz 13489 df-fzo 13632 df-hash 14295 df-word 14469 df-concat 14525 df-s1 14550 df-s2 14803 df-s3 14804 df-trkgc 27954 df-trkgb 27955 df-trkgcb 27956 df-trkg 27959 df-cgrg 28017 df-leg 28089

This theorem is referenced by: legso 28105 krippen 28197 midex 28243 opphllem5 28257 opphllem6 28258

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