Theorem foot 28748

Description: From a point 𝐶 outside of a line 𝐴, there exists a unique point 𝑥 on 𝐴 such that (𝐶𝐿𝑥) is perpendicular to 𝐴. That point is called the foot from 𝐶 on 𝐴. Theorem 8.18 of [Schwabhauser] p. 60. (Contributed by Thierry Arnoux, 19-Oct-2019.)

Hypotheses

Ref	Expression
isperp.p	⊢ 𝑃 = (Base‘𝐺)
isperp.d	⊢ − = (dist‘𝐺)
isperp.i	⊢ 𝐼 = (Itv‘𝐺)
isperp.l	⊢ 𝐿 = (LineG‘𝐺)
isperp.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
isperp.a	⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
foot.x	⊢ (𝜑 → 𝐶 ∈ 𝑃)
foot.y	⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)

Assertion

Ref	Expression
foot	⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐺   𝜑,𝑥   𝑥,𝐶   𝑥,𝐼   𝑥, −   𝑥,𝐿   𝑥,𝑃

Proof of Theorem foot

Dummy variables 𝑢 𝑣 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isperp.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		isperp.d	. . 3 ⊢ − = (dist‘𝐺)
3		isperp.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		isperp.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
5		isperp.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6		isperp.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿)
7		foot.x	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
8		foot.y	. . 3 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)
9	1, 2, 3, 4, 5, 6, 7, 8	footex 28747	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
10		eqid 2740	. . . . . 6 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
11	5	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐺 ∈ TarskiG)
12	7	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶 ∈ 𝑃)
13	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐺 ∈ TarskiG)
14	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐴 ∈ ran 𝐿)
15		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
16	1, 4, 3, 13, 14, 15	tglnpt 28575	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ∈ 𝑃)
17	16	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥 ∈ 𝑃)
18		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ 𝐴)
19	1, 4, 3, 13, 14, 18	tglnpt 28575	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ 𝑃)
20	19	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑧 ∈ 𝑃)
21	8	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ¬ 𝐶 ∈ 𝐴)
22		nelne2 3046	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝑥 ≠ 𝐶)
23	15, 21, 22	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ≠ 𝐶)
24	23	necomd 3002	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐶 ≠ 𝑥)
25	24	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶 ≠ 𝑥)
26	1, 3, 4, 11, 12, 17, 25	tglinerflx1 28659	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶 ∈ (𝐶𝐿𝑥))
27	18	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑧 ∈ 𝐴)
28		simprl 770	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
29	7	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐶 ∈ 𝑃)
30	1, 3, 4, 13, 29, 16, 24	tgelrnln 28656	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐶𝐿𝑥) ∈ ran 𝐿)
31	1, 3, 4, 13, 29, 16, 24	tglinerflx2 28660	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ∈ (𝐶𝐿𝑥))
32	31, 15	elind 4223	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ∈ ((𝐶𝐿𝑥) ∩ 𝐴))
33	1, 2, 3, 4, 13, 30, 14, 32	isperp2 28741	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣 ∈ 𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
34	33	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣 ∈ 𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
35	28, 34	mpbid 232	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣 ∈ 𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺))
36		id 22	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → 𝑢 = 𝐶)
37		eqidd 2741	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → 𝑥 = 𝑥)
38		eqidd 2741	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → 𝑣 = 𝑣)
39	36, 37, 38	s3eqd 14913	. . . . . . . . 9 ⊢ (𝑢 = 𝐶 → ⟨“𝑢𝑥𝑣”⟩ = ⟨“𝐶𝑥𝑣”⟩)
40	39	eleq1d 2829	. . . . . . . 8 ⊢ (𝑢 = 𝐶 → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
41		eqidd 2741	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → 𝐶 = 𝐶)
42		eqidd 2741	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → 𝑥 = 𝑥)
43		id 22	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → 𝑣 = 𝑧)
44	41, 42, 43	s3eqd 14913	. . . . . . . . 9 ⊢ (𝑣 = 𝑧 → ⟨“𝐶𝑥𝑣”⟩ = ⟨“𝐶𝑥𝑧”⟩)
45	44	eleq1d 2829	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → (⟨“𝐶𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺)))
46	40, 45	rspc2va 3647	. . . . . . 7 ⊢ (((𝐶 ∈ (𝐶𝐿𝑥) ∧ 𝑧 ∈ 𝐴) ∧ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣 ∈ 𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺))
47	26, 27, 35, 46	syl21anc 837	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺))
48		nelne2 3046	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝑧 ≠ 𝐶)
49	18, 21, 48	syl2anc 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ≠ 𝐶)
50	49	necomd 3002	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐶 ≠ 𝑧)
51	50	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶 ≠ 𝑧)
52	1, 3, 4, 11, 12, 20, 51	tglinerflx1 28659	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶 ∈ (𝐶𝐿𝑧))
53	15	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥 ∈ 𝐴)
54		simprr 772	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)
55	1, 3, 4, 13, 29, 19, 50	tgelrnln 28656	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐶𝐿𝑧) ∈ ran 𝐿)
56	1, 3, 4, 13, 29, 19, 50	tglinerflx2 28660	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ (𝐶𝐿𝑧))
57	56, 18	elind 4223	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ ((𝐶𝐿𝑧) ∩ 𝐴))
58	1, 2, 3, 4, 13, 55, 14, 57	isperp2 28741	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐶𝐿𝑧)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣 ∈ 𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
59	58	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ((𝐶𝐿𝑧)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣 ∈ 𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
60	54, 59	mpbid 232	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣 ∈ 𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺))
61		eqidd 2741	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → 𝑧 = 𝑧)
62	36, 61, 38	s3eqd 14913	. . . . . . . . 9 ⊢ (𝑢 = 𝐶 → ⟨“𝑢𝑧𝑣”⟩ = ⟨“𝐶𝑧𝑣”⟩)
63	62	eleq1d 2829	. . . . . . . 8 ⊢ (𝑢 = 𝐶 → (⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
64		eqidd 2741	. . . . . . . . . 10 ⊢ (𝑣 = 𝑥 → 𝐶 = 𝐶)
65		eqidd 2741	. . . . . . . . . 10 ⊢ (𝑣 = 𝑥 → 𝑧 = 𝑧)
66		id 22	. . . . . . . . . 10 ⊢ (𝑣 = 𝑥 → 𝑣 = 𝑥)
67	64, 65, 66	s3eqd 14913	. . . . . . . . 9 ⊢ (𝑣 = 𝑥 → ⟨“𝐶𝑧𝑣”⟩ = ⟨“𝐶𝑧𝑥”⟩)
68	67	eleq1d 2829	. . . . . . . 8 ⊢ (𝑣 = 𝑥 → (⟨“𝐶𝑧𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺)))
69	63, 68	rspc2va 3647	. . . . . . 7 ⊢ (((𝐶 ∈ (𝐶𝐿𝑧) ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣 ∈ 𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺))
70	52, 53, 60, 69	syl21anc 837	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺))
71	1, 2, 3, 4, 10, 11, 12, 17, 20, 47, 70	ragflat 28730	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥 = 𝑧)
72	71	ex 412	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
73	72	ralrimivva 3208	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
74		oveq2 7456	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝐶𝐿𝑥) = (𝐶𝐿𝑧))
75	74	breq1d 5176	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴))
76	75	rmo4 3752	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
77	73, 76	sylibr 234	. 2 ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
78		reu5 3390	. 2 ⊢ (∃!𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (∃𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ ∃*𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴))
79	9, 77, 78	sylanbrc 582	1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076  ∃!wreu 3386  ∃*wrmo 3387   class class class wbr 5166  ran crn 5701  ‘cfv 6573  (class class class)co 7448  ⟨“cs3 14891  Basecbs 17258  distcds 17320  TarskiGcstrkg 28453  Itvcitv 28459  LineGclng 28460  pInvGcmir 28678  ∟Gcrag 28719  ⟂Gcperpg 28721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-er 8763  df-map 8886  df-pm 8887  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-dju 9970  df-card 10008  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-2 12356  df-3 12357  df-n0 12554  df-xnn0 12626  df-z 12640  df-uz 12904  df-fz 13568  df-fzo 13712  df-hash 14380  df-word 14563  df-concat 14619  df-s1 14644  df-s2 14897  df-s3 14898  df-trkgc 28474  df-trkgb 28475  df-trkgcb 28476  df-trkg 28479  df-cgrg 28537  df-leg 28609  df-mir 28679  df-rag 28720  df-perpg 28722

This theorem is referenced by:  footeq  28750  mideulem2  28760  lmieu  28810