Theorem foot 28806

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 28805	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
10		eqid 2737	. . . . . 6 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
11	5	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐺 ∈ TarskiG)
12	7	ad2antrr 727	. . . . . 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 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ∈ 𝐴)
16	1, 4, 3, 13, 14, 15	tglnpt 28633	. . . . . . 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 773	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ 𝐴)
19	1, 4, 3, 13, 14, 18	tglnpt 28633	. . . . . . 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 3031	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝑥 ≠ 𝐶)
23	15, 21, 22	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ≠ 𝐶)
24	23	necomd 2988	. . . . . . . . 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 28717	. . . . . . 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 771	. . . . . . . 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 28714	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐶𝐿𝑥) ∈ ran 𝐿)
31	1, 3, 4, 13, 29, 16, 24	tglinerflx2 28718	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ∈ (𝐶𝐿𝑥))
32	31, 15	elind 4154	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ∈ ((𝐶𝐿𝑥) ∩ 𝐴))
33	1, 2, 3, 4, 13, 30, 14, 32	isperp2 28799	. . . . . . . . 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 2738	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → 𝑥 = 𝑥)
38		eqidd 2738	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → 𝑣 = 𝑣)
39	36, 37, 38	s3eqd 14799	. . . . . . . . 9 ⊢ (𝑢 = 𝐶 → ⟨“𝑢𝑥𝑣”⟩ = ⟨“𝐶𝑥𝑣”⟩)
40	39	eleq1d 2822	. . . . . . . 8 ⊢ (𝑢 = 𝐶 → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
41		eqidd 2738	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → 𝐶 = 𝐶)
42		eqidd 2738	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → 𝑥 = 𝑥)
43		id 22	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → 𝑣 = 𝑧)
44	41, 42, 43	s3eqd 14799	. . . . . . . . 9 ⊢ (𝑣 = 𝑧 → ⟨“𝐶𝑥𝑣”⟩ = ⟨“𝐶𝑥𝑧”⟩)
45	44	eleq1d 2822	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → (⟨“𝐶𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺)))
46	40, 45	rspc2va 3590	. . . . . . 7 ⊢ (((𝐶 ∈ (𝐶𝐿𝑥) ∧ 𝑧 ∈ 𝐴) ∧ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣 ∈ 𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺))
47	26, 27, 35, 46	syl21anc 838	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺))
48		nelne2 3031	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝑧 ≠ 𝐶)
49	18, 21, 48	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ≠ 𝐶)
50	49	necomd 2988	. . . . . . . . 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 28717	. . . . . . 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 773	. . . . . . . 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 28714	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐶𝐿𝑧) ∈ ran 𝐿)
56	1, 3, 4, 13, 29, 19, 50	tglinerflx2 28718	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ (𝐶𝐿𝑧))
57	56, 18	elind 4154	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ ((𝐶𝐿𝑧) ∩ 𝐴))
58	1, 2, 3, 4, 13, 55, 14, 57	isperp2 28799	. . . . . . . . 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 2738	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → 𝑧 = 𝑧)
62	36, 61, 38	s3eqd 14799	. . . . . . . . 9 ⊢ (𝑢 = 𝐶 → ⟨“𝑢𝑧𝑣”⟩ = ⟨“𝐶𝑧𝑣”⟩)
63	62	eleq1d 2822	. . . . . . . 8 ⊢ (𝑢 = 𝐶 → (⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
64		eqidd 2738	. . . . . . . . . 10 ⊢ (𝑣 = 𝑥 → 𝐶 = 𝐶)
65		eqidd 2738	. . . . . . . . . 10 ⊢ (𝑣 = 𝑥 → 𝑧 = 𝑧)
66		id 22	. . . . . . . . . 10 ⊢ (𝑣 = 𝑥 → 𝑣 = 𝑥)
67	64, 65, 66	s3eqd 14799	. . . . . . . . 9 ⊢ (𝑣 = 𝑥 → ⟨“𝐶𝑧𝑣”⟩ = ⟨“𝐶𝑧𝑥”⟩)
68	67	eleq1d 2822	. . . . . . . 8 ⊢ (𝑣 = 𝑥 → (⟨“𝐶𝑧𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺)))
69	63, 68	rspc2va 3590	. . . . . . 7 ⊢ (((𝐶 ∈ (𝐶𝐿𝑧) ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣 ∈ 𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)) → ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺))
70	52, 53, 60, 69	syl21anc 838	. . . . . 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 28788	. . . . 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 3181	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
74		oveq2 7376	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝐶𝐿𝑥) = (𝐶𝐿𝑧))
75	74	breq1d 5110	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴))
76	75	rmo4 3690	. . 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 3354	. 2 ⊢ (∃!𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (∃𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ ∃*𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴))
79	9, 77, 78	sylanbrc 584	1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  ∃!wreu 3350  ∃*wrmo 3351   class class class wbr 5100  ran crn 5633  ‘cfv 6500  (class class class)co 7368  ⟨“cs3 14777  Basecbs 17148  distcds 17198  TarskiGcstrkg 28511  Itvcitv 28517  LineGclng 28518  pInvGcmir 28736  ∟Gcrag 28777  ⟂Gcperpg 28779

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-er 8645  df-map 8777  df-pm 8778  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-dju 9825  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-n0 12414  df-xnn0 12487  df-z 12501  df-uz 12764  df-fz 13436  df-fzo 13583  df-hash 14266  df-word 14449  df-concat 14506  df-s1 14532  df-s2 14783  df-s3 14784  df-trkgc 28532  df-trkgb 28533  df-trkgcb 28534  df-trkg 28537  df-cgrg 28595  df-leg 28667  df-mir 28737  df-rag 28778  df-perpg 28780

This theorem is referenced by:  footeq  28808  mideulem2  28818  lmieu  28868