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Theorem foot 27953

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 27952	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
10		eqid 2733	. . . . . 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 482	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐺 ∈ TarskiG)
14	6	adantr 482	. . . . . . . 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 27780	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ∈ 𝑃)
17	16	adantr 482	. . . . . 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 27780	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ 𝑃)
20	19	adantr 482	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑧 ∈ 𝑃)
21	8	adantr 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ¬ 𝐶 ∈ 𝐴)
22		nelne2 3041	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝑥 ≠ 𝐶)
23	15, 21, 22	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ≠ 𝐶)
24	23	necomd 2997	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐶 ≠ 𝑥)
25	24	adantr 482	. . . . . . . 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 27864	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶 ∈ (𝐶𝐿𝑥))
27	18	adantr 482	. . . . . . 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 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐶 ∈ 𝑃)
30	1, 3, 4, 13, 29, 16, 24	tgelrnln 27861	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐶𝐿𝑥) ∈ ran 𝐿)
31	1, 3, 4, 13, 29, 16, 24	tglinerflx2 27865	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ∈ (𝐶𝐿𝑥))
32	31, 15	elind 4193	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑥 ∈ ((𝐶𝐿𝑥) ∩ 𝐴))
33	1, 2, 3, 4, 13, 30, 14, 32	isperp2 27946	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑥)∀𝑣 ∈ 𝐴 ⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
34	33	adantr 482	. . . . . . . 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 231	. . . . . . 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 2734	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → 𝑥 = 𝑥)
38		eqidd 2734	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → 𝑣 = 𝑣)
39	36, 37, 38	s3eqd 14811	. . . . . . . . 9 ⊢ (𝑢 = 𝐶 → ⟨“𝑢𝑥𝑣”⟩ = ⟨“𝐶𝑥𝑣”⟩)
40	39	eleq1d 2819	. . . . . . . 8 ⊢ (𝑢 = 𝐶 → (⟨“𝑢𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑥𝑣”⟩ ∈ (∟G‘𝐺)))
41		eqidd 2734	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → 𝐶 = 𝐶)
42		eqidd 2734	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → 𝑥 = 𝑥)
43		id 22	. . . . . . . . . 10 ⊢ (𝑣 = 𝑧 → 𝑣 = 𝑧)
44	41, 42, 43	s3eqd 14811	. . . . . . . . 9 ⊢ (𝑣 = 𝑧 → ⟨“𝐶𝑥𝑣”⟩ = ⟨“𝐶𝑥𝑧”⟩)
45	44	eleq1d 2819	. . . . . . . 8 ⊢ (𝑣 = 𝑧 → (⟨“𝐶𝑥𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑥𝑧”⟩ ∈ (∟G‘𝐺)))
46	40, 45	rspc2va 3622	. . . . . . 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 3041	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ 𝐴 ∧ ¬ 𝐶 ∈ 𝐴) → 𝑧 ≠ 𝐶)
49	18, 21, 48	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ≠ 𝐶)
50	49	necomd 2997	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐶 ≠ 𝑧)
51	50	adantr 482	. . . . . . . 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 27864	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝐶 ∈ (𝐶𝐿𝑧))
53	15	adantr 482	. . . . . . 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 27861	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐶𝐿𝑧) ∈ ran 𝐿)
56	1, 3, 4, 13, 29, 19, 50	tglinerflx2 27865	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ (𝐶𝐿𝑧))
57	56, 18	elind 4193	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑧 ∈ ((𝐶𝐿𝑧) ∩ 𝐴))
58	1, 2, 3, 4, 13, 55, 14, 57	isperp2 27946	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐶𝐿𝑧)(⟂G‘𝐺)𝐴 ↔ ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣 ∈ 𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
59	58	adantr 482	. . . . . . . 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 231	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → ∀𝑢 ∈ (𝐶𝐿𝑧)∀𝑣 ∈ 𝐴 ⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺))
61		eqidd 2734	. . . . . . . . . 10 ⊢ (𝑢 = 𝐶 → 𝑧 = 𝑧)
62	36, 61, 38	s3eqd 14811	. . . . . . . . 9 ⊢ (𝑢 = 𝐶 → ⟨“𝑢𝑧𝑣”⟩ = ⟨“𝐶𝑧𝑣”⟩)
63	62	eleq1d 2819	. . . . . . . 8 ⊢ (𝑢 = 𝐶 → (⟨“𝑢𝑧𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑧𝑣”⟩ ∈ (∟G‘𝐺)))
64		eqidd 2734	. . . . . . . . . 10 ⊢ (𝑣 = 𝑥 → 𝐶 = 𝐶)
65		eqidd 2734	. . . . . . . . . 10 ⊢ (𝑣 = 𝑥 → 𝑧 = 𝑧)
66		id 22	. . . . . . . . . 10 ⊢ (𝑣 = 𝑥 → 𝑣 = 𝑥)
67	64, 65, 66	s3eqd 14811	. . . . . . . . 9 ⊢ (𝑣 = 𝑥 → ⟨“𝐶𝑧𝑣”⟩ = ⟨“𝐶𝑧𝑥”⟩)
68	67	eleq1d 2819	. . . . . . . 8 ⊢ (𝑣 = 𝑥 → (⟨“𝐶𝑧𝑣”⟩ ∈ (∟G‘𝐺) ↔ ⟨“𝐶𝑧𝑥”⟩ ∈ (∟G‘𝐺)))
69	63, 68	rspc2va 3622	. . . . . . 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 27935	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴)) → 𝑥 = 𝑧)
72	71	ex 414	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
73	72	ralrimivva 3201	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ∧ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴) → 𝑥 = 𝑧))
74		oveq2 7412	. . . . 5 ⊢ (𝑥 = 𝑧 → (𝐶𝐿𝑥) = (𝐶𝐿𝑧))
75	74	breq1d 5157	. . . 4 ⊢ (𝑥 = 𝑧 → ((𝐶𝐿𝑥)(⟂G‘𝐺)𝐴 ↔ (𝐶𝐿𝑧)(⟂G‘𝐺)𝐴))
76	75	rmo4 3725	. . 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 233	. 2 ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (𝐶𝐿𝑥)(⟂G‘𝐺)𝐴)
78		reu5 3379	. 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)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ∃!wreu 3375 ∃*wrmo 3376 class class class wbr 5147 ran crn 5676 ‘cfv 6540 (class class class)co 7404 ⟨“cs3 14789 Basecbs 17140 distcds 17202 TarskiGcstrkg 27658 Itvcitv 27664 LineGclng 27665 pInvGcmir 27883 ∟Gcrag 27924 ⟂Gcperpg 27926

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-oadd 8465 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-concat 14517 df-s1 14542 df-s2 14795 df-s3 14796 df-trkgc 27679 df-trkgb 27680 df-trkgcb 27681 df-trkg 27684 df-cgrg 27742 df-leg 27814 df-mir 27884 df-rag 27925 df-perpg 27927

This theorem is referenced by: footeq 27955 mideulem2 27965 lmieu 28015

W3C validator