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Theorem rrxdstprj1 25619

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 787	. . 3 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) → 𝐼 ∈ 𝑉)
2		simpr 490	. . 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 781	. . 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 783	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐹 ∈ 𝑋)
6	4, 5	rrxfsupp 25612	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹 supp 0) ∈ Fin)
7		simprr 785	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐺 ∈ 𝑋)
8	4, 7	rrxfsupp 25612	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺 supp 0) ∈ Fin)
9		unfi 9162	. . . . . . . 8 ⊢ (((𝐹 supp 0) ∈ Fin ∧ (𝐺 supp 0) ∈ Fin) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ∈ Fin)
10	6, 8, 9	syl2anc 596	. . . . . . 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 25613	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹 supp 0) ⊆ 𝐼)
12	4, 7	rrxsuppss 25613	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺 supp 0) ⊆ 𝐼)
13	11, 12	unssd 4145	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐼)
14	13	sselda 3938	. . . . . . . 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 25611	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐹:𝐼⟶ℝ)
16	15	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ ℝ)
17	4, 7	rrxf 25611	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐺:𝐼⟶ℝ)
18	17	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝐺‘𝑘) ∈ ℝ)
19	16, 18	resubcld 11657	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℝ)
20	19	resqcld 14179	. . . . . . . 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 603	. . . . . . 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 14191	. . . . . . . 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 603	. . . . . . 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 6885	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (𝐹‘𝑘) = (𝐹‘𝐴))
25		fveq2 6885	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (𝐺‘𝑘) = (𝐺‘𝐴))
26	24, 25	oveq12d 7437	. . . . . . . 8 ⊢ (𝑘 = 𝐴 → ((𝐹‘𝑘) − (𝐺‘𝑘)) = ((𝐹‘𝐴) − (𝐺‘𝐴)))
27	26	oveq1d 7434	. . . . . . 7 ⊢ (𝑘 = 𝐴 → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) = (((𝐹‘𝐴) − (𝐺‘𝐴))↑2))
28		simplr 781	. . . . . . 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 15872	. . . . . 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 3939	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐴 ∈ 𝐼)
31	15, 30	ffvelcdmd 7084	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹‘𝐴) ∈ ℝ)
32	17, 30	ffvelcdmd 7084	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺‘𝐴) ∈ ℝ)
33	31, 32	resubcld 11657	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴) − (𝐺‘𝐴)) ∈ ℝ)
34		absresq 15377	. . . . . . 7 ⊢ (((𝐹‘𝐴) − (𝐺‘𝐴)) ∈ ℝ → ((abs‘((𝐹‘𝐴) − (𝐺‘𝐴)))↑2) = (((𝐹‘𝐴) − (𝐺‘𝐴))↑2))
35	33, 34	syl 18	. . . . . 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 15808	. . . . . . 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 15870	. . . . . . 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 15330	. . . . . . 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 596	. . . . . 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 5145	. . . . 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 11252	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴) − (𝐺‘𝐴)) ∈ ℂ)
42	41	abscld 15514	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (abs‘((𝐹‘𝐴) − (𝐺‘𝐴))) ∈ ℝ)
43	36, 37	resqrtcld 15493	. . . . . 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 15522	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 0 ≤ (abs‘((𝐹‘𝐴) − (𝐺‘𝐴))))
45	36, 37	sqrtge0d 15496	. . . . . 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 14311	. . . . 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 260	. . . 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 24997	. . . . 5 ⊢ (((𝐹‘𝐴) ∈ ℝ ∧ (𝐺‘𝐴) ∈ ℝ) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) = (abs‘((𝐹‘𝐴) − (𝐺‘𝐴))))
50	31, 32, 49	syl2anc 596	. . . 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 25615	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
53	52	3expb 1138	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
54	53	adantlr 728	. . . 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 5145	. . 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 851	. 2 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) ≤ (𝐹𝐷𝐺))
57		simplll 787	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐼 ∈ 𝑉)
58		simplrl 789	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐹 ∈ 𝑋)
59		ssun1 4131	. . . . . . . . . 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 4100	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ⊆ (𝐼 ∖ (𝐹 supp 0)))
62	61	sselda 3938	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐴 ∈ (𝐼 ∖ (𝐹 supp 0)))
63		simpr 490	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 𝐹 ∈ 𝑋)
64	4, 63	rrxf 25611	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 𝐹:𝐼⟶ℝ)
65		ssidd 3961	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → (𝐹 supp 0) ⊆ (𝐹 supp 0))
66		simpl 488	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 𝐼 ∈ 𝑉)
67		0red 11226	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 0 ∈ ℝ)
68	64, 65, 66, 67	suppssr 8197	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) ∧ 𝐴 ∈ (𝐼 ∖ (𝐹 supp 0))) → (𝐹‘𝐴) = 0)
69	57, 58, 62, 68	syl21anc 851	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐹‘𝐴) = 0)
70		0red 11226	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 0 ∈ ℝ)
71	69, 70	eqeltrd 2865	. . . . 5 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐹‘𝐴) ∈ ℝ)
72		simplrr 790	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐺 ∈ 𝑋)
73		ssun2 4132	. . . . . . . . . 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 4100	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ⊆ (𝐼 ∖ (𝐺 supp 0)))
76	75	sselda 3938	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐴 ∈ (𝐼 ∖ (𝐺 supp 0)))
77		simpr 490	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 𝐺 ∈ 𝑋)
78	4, 77	rrxf 25611	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 𝐺:𝐼⟶ℝ)
79		ssidd 3961	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → (𝐺 supp 0) ⊆ (𝐺 supp 0))
80		simpl 488	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 𝐼 ∈ 𝑉)
81		0red 11226	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 0 ∈ ℝ)
82	78, 79, 80, 81	suppssr 8197	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) ∧ 𝐴 ∈ (𝐼 ∖ (𝐺 supp 0))) → (𝐺‘𝐴) = 0)
83	57, 72, 76, 82	syl21anc 851	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐺‘𝐴) = 0)
84	83, 70	eqeltrd 2865	. . . . 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 596	. . . 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 7437	. . . . . 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 12374	. . . . . 6 ⊢ (0 − 0) = 0
88	86, 87	eqtrdi 2816	. . . . 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 15366	. . . 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 2800	. . 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 25618	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → 𝐷 ∈ (Met‘𝑋))
92	91	ad3antrrr 743	. . . 4 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐷 ∈ (Met‘𝑋))
93		metge0 24553	. . . 4 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 0 ≤ (𝐹𝐷𝐺))
94	92, 58, 72, 93	syl3anc 1398	. . 3 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 0 ≤ (𝐹𝐷𝐺))
95	90, 94	eqbrtrd 5135	. 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 781	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐴 ∈ 𝐼)
97		simprl 783	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐹 ∈ 𝑋)
98	4, 97	rrxsuppss 25613	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹 supp 0) ⊆ 𝐼)
99		simprr 785	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐺 ∈ 𝑋)
100	4, 99	rrxsuppss 25613	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺 supp 0) ⊆ 𝐼)
101	98, 100	unssd 4145	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐼)
102		undif 4445	. . . . 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 221	. . . 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 2868	. . 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 4107	. . 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 221	. 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 973	1 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) ≤ (𝐹𝐷𝐺))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2146 {crab 3418 ∖ cdif 3903 ∪ cun 3904 ⊆ wss 3906 class class class wbr 5111 × cxp 5661 ↾ cres 5665 ∘ ccom 5667 ‘cfv 6540 (class class class)co 7419 supp csupp 8162 ↑_m cmap 8830 Fincfn 8949 finSupp cfsupp 9328 ℝcr 11114 0cc0 11115 ≤ cle 11259 − cmin 11456 2c2 12310 ↑cexp 14115 √csqrt 15308 abscabs 15309 Σcsu 15761 distcds 17341 Metcmet 21558 ℝ^crrx 25593

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193 ax-addf 11194 ax-mulf 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-tpos 8228 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-ixp 8902 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-sup 9409 df-oi 9479 df-card 9941 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-z 12607 df-dec 12728 df-uz 12879 df-rp 13033 df-xneg 13153 df-xadd 13154 df-xmul 13155 df-ico 13394 df-fz 13552 df-fzo 13700 df-seq 14056 df-exp 14116 df-hash 14385 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-clim 15563 df-sum 15762 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-mulr 17346 df-starv 17347 df-sca 17348 df-vsca 17349 df-ip 17350 df-tset 17351 df-ple 17352 df-ds 17354 df-unif 17355 df-hom 17356 df-cco 17357 df-0g 17516 df-gsum 17517 df-prds 17522 df-pws 17524 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-mhm 18878 df-grp 19047 df-minusg 19048 df-sbg 19049 df-subg 19233 df-ghm 19328 df-cntz 19431 df-cmn 19896 df-abl 19897 df-mgp 20261 df-rng 20275 df-ur 20308 df-ring 20361 df-cring 20362 df-oppr 20465 df-dvdsr 20485 df-unit 20486 df-invr 20516 df-dvr 20529 df-rhm 20600 df-subrng 20695 df-subrg 20719 df-drng 20879 df-field 20880 df-staf 20992 df-srng 20993 df-lmod 21033 df-lss 21103 df-sra 21344 df-rgmod 21345 df-xmet 21565 df-met 21566 df-cnfld 21573 df-refld 21805 df-dsmm 21932 df-frlm 21947 df-nm 24790 df-tng 24792 df-tcph 25379 df-rrx 25595

This theorem is used by: rrnprjdstle 47073

W3C validator