Theorem foot 26516

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 26515	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
10		eqid 2798	. . . . . 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 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐺 ∈ TarskiG)
14	6	adantr 484	. . . . . . . 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 26343	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ∈ 𝑃)
17	16	adantr 484	. . . . . 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 26343	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ 𝑃)
20	19	adantr 484	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑧 ∈ 𝑃)
21	8	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ¬ 𝐶 ∈ 𝐴)
22		nelne2 3084	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝑥 ≠ 𝐶)
23	15, 21, 22	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ≠ 𝐶)
24	23	necomd 3042	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐶 ≠ 𝑥)
25	24	adantr 484	. . . . . . . 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 26427	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶 ∈ (𝐶𝐿𝑥))
27	18	adantr 484	. . . . . . 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 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐶 ∈ 𝑃)
30	1, 3, 4, 13, 29, 16, 24	tgelrnln 26424	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐶𝐿𝑥) ∈ ran 𝐿)
31	1, 3, 4, 13, 29, 16, 24	tglinerflx2 26428	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ∈ (𝐶𝐿𝑥))
32	31, 15	elind 4121	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ∈ ((𝐶𝐿𝑥) ∩ 𝐴))
33	1, 2, 3, 4, 13, 30, 14, 32	isperp2 26509	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣 ∈ 𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
34	33	adantr 484	. . . . . . . 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 235	. . . . . . 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 2799	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → 𝑥 = 𝑥)
38		eqidd 2799	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → 𝑣 = 𝑣)
39	36, 37, 38	s3eqd 14217	. . . . . . . . 9 ⊢ (𝑢 = 𝐶 → ⟨“𝑢𝑥𝑣”⟩ = ⟨“𝐶𝑥𝑣”⟩)
40	39	eleq1d 2874	. . . . . . . 8 ⊢ (𝑢 = 𝐶 → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
41		eqidd 2799	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → 𝐶 = 𝐶)
42		eqidd 2799	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → 𝑥 = 𝑥)
43		id 22	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → 𝑣 = 𝑧)
44	41, 42, 43	s3eqd 14217	. . . . . . . . 9 ⊢ (𝑣 = 𝑧 → ⟨“𝐶𝑥𝑣”⟩ = ⟨“𝐶𝑥𝑧”⟩)
45	44	eleq1d 2874	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → (⟨“𝐶𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺)))
46	40, 45	rspc2va 3582	. . . . . . 7 ⊢ (((𝐶 ∈ (𝐶𝐿𝑥) ∧ 𝑧 ∈ 𝐴) ∧ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣 ∈ 𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺))
47	26, 27, 35, 46	syl21anc 836	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺))
48		nelne2 3084	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝑧 ≠ 𝐶)
49	18, 21, 48	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ≠ 𝐶)
50	49	necomd 3042	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐶 ≠ 𝑧)
51	50	adantr 484	. . . . . . . 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 26427	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶 ∈ (𝐶𝐿𝑧))
53	15	adantr 484	. . . . . . 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 26424	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐶𝐿𝑧) ∈ ran 𝐿)
56	1, 3, 4, 13, 29, 19, 50	tglinerflx2 26428	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ (𝐶𝐿𝑧))
57	56, 18	elind 4121	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ ((𝐶𝐿𝑧) ∩ 𝐴))
58	1, 2, 3, 4, 13, 55, 14, 57	isperp2 26509	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐶𝐿𝑧)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣 ∈ 𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
59	58	adantr 484	. . . . . . . 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 235	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣 ∈ 𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺))
61		eqidd 2799	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → 𝑧 = 𝑧)
62	36, 61, 38	s3eqd 14217	. . . . . . . . 9 ⊢ (𝑢 = 𝐶 → ⟨“𝑢𝑧𝑣”⟩ = ⟨“𝐶𝑧𝑣”⟩)
63	62	eleq1d 2874	. . . . . . . 8 ⊢ (𝑢 = 𝐶 → (⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
64		eqidd 2799	. . . . . . . . . 10 ⊢ (𝑣 = 𝑥 → 𝐶 = 𝐶)
65		eqidd 2799	. . . . . . . . . 10 ⊢ (𝑣 = 𝑥 → 𝑧 = 𝑧)
66		id 22	. . . . . . . . . 10 ⊢ (𝑣 = 𝑥 → 𝑣 = 𝑥)
67	64, 65, 66	s3eqd 14217	. . . . . . . . 9 ⊢ (𝑣 = 𝑥 → ⟨“𝐶𝑧𝑣”⟩ = ⟨“𝐶𝑧𝑥”⟩)
68	67	eleq1d 2874	. . . . . . . 8 ⊢ (𝑣 = 𝑥 → (⟨“𝐶𝑧𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺)))
69	63, 68	rspc2va 3582	. . . . . . 7 ⊢ (((𝐶 ∈ (𝐶𝐿𝑧) ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣 ∈ 𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺))
70	52, 53, 60, 69	syl21anc 836	. . . . . 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 26498	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥 = 𝑧)
72	71	ex 416	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
73	72	ralrimivva 3156	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
74		oveq2 7143	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝐶𝐿𝑥) = (𝐶𝐿𝑧))
75	74	breq1d 5040	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴))
76	75	rmo4 3669	. . 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 237	. 2 ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
78		reu5 3375	. 2 ⊢ (∃!𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (∃𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ ∃*𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴))
79	9, 77, 78	sylanbrc 586	1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  ∃!wreu 3108  ∃*wrmo 3109   class class class wbr 5030  ran crn 5520  ‘cfv 6324  (class class class)co 7135  ⟨“cs3 14195  Basecbs 16475  distcds 16566  TarskiGcstrkg 26224  Itvcitv 26230  LineGclng 26231  pInvGcmir 26446  ∟Gcrag 26487  ⟂Gcperpg 26489

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-dju 9314  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-hash 13687  df-word 13858  df-concat 13914  df-s1 13941  df-s2 14201  df-s3 14202  df-trkgc 26242  df-trkgb 26243  df-trkgcb 26244  df-trkg 26247  df-cgrg 26305  df-leg 26377  df-mir 26447  df-rag 26488  df-perpg 26490

This theorem is referenced by:  footeq  26518  mideulem2  26528  lmieu  26578