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Theorem rrxdstprj1 25307

Description: The distance between two points in Euclidean space is greater than the distance between the projections onto one coordinate. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.) (Revised by Thierry Arnoux, 7-Jul-2019.)

**Hypotheses**
Ref	Expression
rrxmval.1	⊢ 𝑋 = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0}
rrxmval.d	⊢ 𝐷 = (dist‘(ℝ^‘𝐼))
rrxdstprj1.1	⊢ 𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ))

**Assertion**
Ref	Expression
rrxdstprj1	⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) ≤ (𝐹𝐷𝐺))

Distinct variable groups: ℎ,𝐹 ℎ,𝐺 ℎ,𝐼 ℎ,𝑉 Allowed substitution hints: 𝐴(ℎ) 𝐷(ℎ) 𝑀(ℎ) 𝑋(ℎ)

Proof of Theorem rrxdstprj1 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplll 774	. . 3 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) → 𝐼 ∈ 𝑉)
2		simpr 484	. . 3 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) → 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0)))
3		simplr 768	. . 3 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) → (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋))
4		rrxmval.1	. . . . . . . . 9 ⊢ 𝑋 = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0}
5		simprl 770	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐹 ∈ 𝑋)
6	4, 5	rrxfsupp 25300	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹 supp 0) ∈ Fin)
7		simprr 772	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐺 ∈ 𝑋)
8	4, 7	rrxfsupp 25300	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺 supp 0) ∈ Fin)
9		unfi 9085	. . . . . . . 8 ⊢ (((𝐹 supp 0) ∈ Fin ∧ (𝐺 supp 0) ∈ Fin) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ∈ Fin)
10	6, 8, 9	syl2anc 584	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ∈ Fin)
11	4, 5	rrxsuppss 25301	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹 supp 0) ⊆ 𝐼)
12	4, 7	rrxsuppss 25301	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺 supp 0) ⊆ 𝐼)
13	11, 12	unssd 4143	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐼)
14	13	sselda 3935	. . . . . . . 8 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) → 𝑘 ∈ 𝐼)
15	4, 5	rrxf 25299	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐹:𝐼⟶ℝ)
16	15	ffvelcdmda 7018	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ ℝ)
17	4, 7	rrxf 25299	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐺:𝐼⟶ℝ)
18	17	ffvelcdmda 7018	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝐺‘𝑘) ∈ ℝ)
19	16, 18	resubcld 11548	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℝ)
20	19	resqcld 14032	. . . . . . . 8 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) ∈ ℝ)
21	14, 20	syldan 591	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) ∈ ℝ)
22	19	sqge0d 14044	. . . . . . . 8 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → 0 ≤ (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
23	14, 22	syldan 591	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) → 0 ≤ (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
24		fveq2 6822	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (𝐹‘𝑘) = (𝐹‘𝐴))
25		fveq2 6822	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (𝐺‘𝑘) = (𝐺‘𝐴))
26	24, 25	oveq12d 7367	. . . . . . . 8 ⊢ (𝑘 = 𝐴 → ((𝐹‘𝑘) − (𝐺‘𝑘)) = ((𝐹‘𝐴) − (𝐺‘𝐴)))
27	26	oveq1d 7364	. . . . . . 7 ⊢ (𝑘 = 𝐴 → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) = (((𝐹‘𝐴) − (𝐺‘𝐴))↑2))
28		simplr 768	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0)))
29	10, 21, 23, 27, 28	fsumge1 15704	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (((𝐹‘𝐴) − (𝐺‘𝐴))↑2) ≤ Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
30	13, 28	sseldd 3936	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐴 ∈ 𝐼)
31	15, 30	ffvelcdmd 7019	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹‘𝐴) ∈ ℝ)
32	17, 30	ffvelcdmd 7019	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺‘𝐴) ∈ ℝ)
33	31, 32	resubcld 11548	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴) − (𝐺‘𝐴)) ∈ ℝ)
34		absresq 15209	. . . . . . 7 ⊢ (((𝐹‘𝐴) − (𝐺‘𝐴)) ∈ ℝ → ((abs‘((𝐹‘𝐴) − (𝐺‘𝐴)))↑2) = (((𝐹‘𝐴) − (𝐺‘𝐴))↑2))
35	33, 34	syl 17	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((abs‘((𝐹‘𝐴) − (𝐺‘𝐴)))↑2) = (((𝐹‘𝐴) − (𝐺‘𝐴))↑2))
36	10, 21	fsumrecl 15641	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2) ∈ ℝ)
37	10, 21, 23	fsumge0 15702	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 0 ≤ Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
38		resqrtth 15162	. . . . . . 7 ⊢ ((Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2) ∈ ℝ ∧ 0 ≤ Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)) → ((√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2))↑2) = Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
39	36, 37, 38	syl2anc 584	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2))↑2) = Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
40	29, 35, 39	3brtr4d 5124	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((abs‘((𝐹‘𝐴) − (𝐺‘𝐴)))↑2) ≤ ((√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2))↑2))
41	33	recnd 11143	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴) − (𝐺‘𝐴)) ∈ ℂ)
42	41	abscld 15346	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (abs‘((𝐹‘𝐴) − (𝐺‘𝐴))) ∈ ℝ)
43	36, 37	resqrtcld 15325	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)) ∈ ℝ)
44	41	absge0d 15354	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 0 ≤ (abs‘((𝐹‘𝐴) − (𝐺‘𝐴))))
45	36, 37	sqrtge0d 15328	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 0 ≤ (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
46	42, 43, 44, 45	le2sqd 14164	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((abs‘((𝐹‘𝐴) − (𝐺‘𝐴))) ≤ (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)) ↔ ((abs‘((𝐹‘𝐴) − (𝐺‘𝐴)))↑2) ≤ ((√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2))↑2)))
47	40, 46	mpbird 257	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (abs‘((𝐹‘𝐴) − (𝐺‘𝐴))) ≤ (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
48		rrxdstprj1.1	. . . . . 6 ⊢ 𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ))
49	48	remetdval 24675	. . . . 5 ⊢ (((𝐹‘𝐴) ∈ ℝ ∧ (𝐺‘𝐴) ∈ ℝ) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) = (abs‘((𝐹‘𝐴) − (𝐺‘𝐴))))
50	31, 32, 49	syl2anc 584	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) = (abs‘((𝐹‘𝐴) − (𝐺‘𝐴))))
51		rrxmval.d	. . . . . . 7 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))
52	4, 51	rrxmval 25303	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
53	52	3expb 1120	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
54	53	adantlr 715	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
55	47, 50, 54	3brtr4d 5124	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) ≤ (𝐹𝐷𝐺))
56	1, 2, 3, 55	syl21anc 837	. 2 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) ≤ (𝐹𝐷𝐺))
57		simplll 774	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐼 ∈ 𝑉)
58		simplrl 776	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐹 ∈ 𝑋)
59		ssun1 4129	. . . . . . . . . 10 ⊢ (𝐹 supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0))
60	59	a1i 11	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹 supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0)))
61	60	sscond 4097	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ⊆ (𝐼 ∖ (𝐹 supp 0)))
62	61	sselda 3935	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐴 ∈ (𝐼 ∖ (𝐹 supp 0)))
63		simpr 484	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 𝐹 ∈ 𝑋)
64	4, 63	rrxf 25299	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 𝐹:𝐼⟶ℝ)
65		ssidd 3959	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → (𝐹 supp 0) ⊆ (𝐹 supp 0))
66		simpl 482	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 𝐼 ∈ 𝑉)
67		0red 11118	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 0 ∈ ℝ)
68	64, 65, 66, 67	suppssr 8128	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) ∧ 𝐴 ∈ (𝐼 ∖ (𝐹 supp 0))) → (𝐹‘𝐴) = 0)
69	57, 58, 62, 68	syl21anc 837	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐹‘𝐴) = 0)
70		0red 11118	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 0 ∈ ℝ)
71	69, 70	eqeltrd 2828	. . . . 5 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐹‘𝐴) ∈ ℝ)
72		simplrr 777	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐺 ∈ 𝑋)
73		ssun2 4130	. . . . . . . . . 10 ⊢ (𝐺 supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0))
74	73	a1i 11	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺 supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0)))
75	74	sscond 4097	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ⊆ (𝐼 ∖ (𝐺 supp 0)))
76	75	sselda 3935	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐴 ∈ (𝐼 ∖ (𝐺 supp 0)))
77		simpr 484	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 𝐺 ∈ 𝑋)
78	4, 77	rrxf 25299	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 𝐺:𝐼⟶ℝ)
79		ssidd 3959	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → (𝐺 supp 0) ⊆ (𝐺 supp 0))
80		simpl 482	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 𝐼 ∈ 𝑉)
81		0red 11118	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 0 ∈ ℝ)
82	78, 79, 80, 81	suppssr 8128	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) ∧ 𝐴 ∈ (𝐼 ∖ (𝐺 supp 0))) → (𝐺‘𝐴) = 0)
83	57, 72, 76, 82	syl21anc 837	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐺‘𝐴) = 0)
84	83, 70	eqeltrd 2828	. . . . 5 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐺‘𝐴) ∈ ℝ)
85	71, 84, 49	syl2anc 584	. . . 4 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) = (abs‘((𝐹‘𝐴) − (𝐺‘𝐴))))
86	69, 83	oveq12d 7367	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → ((𝐹‘𝐴) − (𝐺‘𝐴)) = (0 − 0))
87		0m0e0 12243	. . . . . 6 ⊢ (0 − 0) = 0
88	86, 87	eqtrdi 2780	. . . . 5 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → ((𝐹‘𝐴) − (𝐺‘𝐴)) = 0)
89	88	abs00bd 15198	. . . 4 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (abs‘((𝐹‘𝐴) − (𝐺‘𝐴))) = 0)
90	85, 89	eqtrd 2764	. . 3 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) = 0)
91	4, 51	rrxmet 25306	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → 𝐷 ∈ (Met‘𝑋))
92	91	ad3antrrr 730	. . . 4 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐷 ∈ (Met‘𝑋))
93		metge0 24231	. . . 4 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 0 ≤ (𝐹𝐷𝐺))
94	92, 58, 72, 93	syl3anc 1373	. . 3 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 0 ≤ (𝐹𝐷𝐺))
95	90, 94	eqbrtrd 5114	. 2 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) ≤ (𝐹𝐷𝐺))
96		simplr 768	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐴 ∈ 𝐼)
97		simprl 770	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐹 ∈ 𝑋)
98	4, 97	rrxsuppss 25301	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹 supp 0) ⊆ 𝐼)
99		simprr 772	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐺 ∈ 𝑋)
100	4, 99	rrxsuppss 25301	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺 supp 0) ⊆ 𝐼)
101	98, 100	unssd 4143	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐼)
102		undif 4433	. . . . 5 ⊢ (((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐼 ↔ (((𝐹 supp 0) ∪ (𝐺 supp 0)) ∪ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) = 𝐼)
103	101, 102	sylib 218	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (((𝐹 supp 0) ∪ (𝐺 supp 0)) ∪ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) = 𝐼)
104	96, 103	eleqtrrd 2831	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐴 ∈ (((𝐹 supp 0) ∪ (𝐺 supp 0)) ∪ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))))
105		elun 4104	. . 3 ⊢ (𝐴 ∈ (((𝐹 supp 0) ∪ (𝐺 supp 0)) ∪ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) ↔ (𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0)) ∨ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))))
106	104, 105	sylib 218	. 2 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0)) ∨ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))))
107	56, 95, 106	mpjaodan 960	1 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) ≤ (𝐹𝐷𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 {crab 3394 ∖ cdif 3900 ∪ cun 3901 ⊆ wss 3903 class class class wbr 5092 × cxp 5617 ↾ cres 5621 ∘ ccom 5623 ‘cfv 6482 (class class class)co 7349 supp csupp 8093 ↑_m cmap 8753 Fincfn 8872 finSupp cfsupp 9251 ℝcr 11008 0cc0 11009 ≤ cle 11150 − cmin 11347 2c2 12183 ↑cexp 13968 √csqrt 15140 abscabs 15141 Σcsu 15593 distcds 17170 Metcmet 21247 ℝ^crrx 25281

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-tpos 8159 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-map 8755 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-sup 9332 df-oi 9402 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ico 13254 df-fz 13411 df-fzo 13558 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-mhm 18657 df-grp 18815 df-minusg 18816 df-sbg 18817 df-subg 19002 df-ghm 19092 df-cntz 19196 df-cmn 19661 df-abl 19662 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-cring 20121 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-invr 20273 df-dvr 20286 df-rhm 20357 df-subrng 20431 df-subrg 20455 df-drng 20616 df-field 20617 df-staf 20724 df-srng 20725 df-lmod 20765 df-lss 20835 df-sra 21077 df-rgmod 21078 df-xmet 21254 df-met 21255 df-cnfld 21262 df-refld 21512 df-dsmm 21639 df-frlm 21654 df-nm 24468 df-tng 24470 df-tcph 25067 df-rrx 25283

This theorem is referenced by: rrnprjdstle 46286

W3C validator