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Theorem rrxdstprj1 24671

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 772	. . 3 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) → 𝐼 ∈ 𝑉)
2		simpr 485	. . 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 766	. . 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 768	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐹 ∈ 𝑋)
6	4, 5	rrxfsupp 24664	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹 supp 0) ∈ Fin)
7		simprr 770	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐺 ∈ 𝑋)
8	4, 7	rrxfsupp 24664	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺 supp 0) ∈ Fin)
9		unfi 9029	. . . . . . . 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 24665	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹 supp 0) ⊆ 𝐼)
12	4, 7	rrxsuppss 24665	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺 supp 0) ⊆ 𝐼)
13	11, 12	unssd 4132	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐼)
14	13	sselda 3931	. . . . . . . 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 24663	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐹:𝐼⟶ℝ)
16	15	ffvelcdmda 7011	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ ℝ)
17	4, 7	rrxf 24663	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐺:𝐼⟶ℝ)
18	17	ffvelcdmda 7011	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝐺‘𝑘) ∈ ℝ)
19	16, 18	resubcld 11496	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℝ)
20	19	resqcld 14058	. . . . . . . 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 14059	. . . . . . . 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 6819	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (𝐹‘𝑘) = (𝐹‘𝐴))
25		fveq2 6819	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (𝐺‘𝑘) = (𝐺‘𝐴))
26	24, 25	oveq12d 7347	. . . . . . . 8 ⊢ (𝑘 = 𝐴 → ((𝐹‘𝑘) − (𝐺‘𝑘)) = ((𝐹‘𝐴) − (𝐺‘𝐴)))
27	26	oveq1d 7344	. . . . . . 7 ⊢ (𝑘 = 𝐴 → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) = (((𝐹‘𝐴) − (𝐺‘𝐴))↑2))
28		simplr 766	. . . . . . 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 15600	. . . . . 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 3932	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐴 ∈ 𝐼)
31	15, 30	ffvelcdmd 7012	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹‘𝐴) ∈ ℝ)
32	17, 30	ffvelcdmd 7012	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺‘𝐴) ∈ ℝ)
33	31, 32	resubcld 11496	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴) − (𝐺‘𝐴)) ∈ ℝ)
34		absresq 15105	. . . . . . 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 15537	. . . . . . 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 15598	. . . . . . 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 15058	. . . . . . 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 5121	. . . . 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 11096	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴) − (𝐺‘𝐴)) ∈ ℂ)
42	41	abscld 15239	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (abs‘((𝐹‘𝐴) − (𝐺‘𝐴))) ∈ ℝ)
43	36, 37	resqrtcld 15220	. . . . . 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 15247	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 0 ≤ (abs‘((𝐹‘𝐴) − (𝐺‘𝐴))))
45	36, 37	sqrtge0d 15223	. . . . . 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 14067	. . . . 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 256	. . . 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 24050	. . . . 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 24667	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
53	52	3expb 1119	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
54	53	adantlr 712	. . . 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 5121	. . 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 835	. 2 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) ≤ (𝐹𝐷𝐺))
57		simplll 772	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐼 ∈ 𝑉)
58		simplrl 774	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐹 ∈ 𝑋)
59		ssun1 4118	. . . . . . . . . 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 4087	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ⊆ (𝐼 ∖ (𝐹 supp 0)))
62	61	sselda 3931	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐴 ∈ (𝐼 ∖ (𝐹 supp 0)))
63		simpr 485	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 𝐹 ∈ 𝑋)
64	4, 63	rrxf 24663	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 𝐹:𝐼⟶ℝ)
65		ssidd 3954	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → (𝐹 supp 0) ⊆ (𝐹 supp 0))
66		simpl 483	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 𝐼 ∈ 𝑉)
67		0red 11071	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 0 ∈ ℝ)
68	64, 65, 66, 67	suppssr 8074	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) ∧ 𝐴 ∈ (𝐼 ∖ (𝐹 supp 0))) → (𝐹‘𝐴) = 0)
69	57, 58, 62, 68	syl21anc 835	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐹‘𝐴) = 0)
70		0red 11071	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 0 ∈ ℝ)
71	69, 70	eqeltrd 2837	. . . . 5 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐹‘𝐴) ∈ ℝ)
72		simplrr 775	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐺 ∈ 𝑋)
73		ssun2 4119	. . . . . . . . . 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 4087	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ⊆ (𝐼 ∖ (𝐺 supp 0)))
76	75	sselda 3931	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐴 ∈ (𝐼 ∖ (𝐺 supp 0)))
77		simpr 485	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 𝐺 ∈ 𝑋)
78	4, 77	rrxf 24663	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 𝐺:𝐼⟶ℝ)
79		ssidd 3954	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → (𝐺 supp 0) ⊆ (𝐺 supp 0))
80		simpl 483	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 𝐼 ∈ 𝑉)
81		0red 11071	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 0 ∈ ℝ)
82	78, 79, 80, 81	suppssr 8074	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) ∧ 𝐴 ∈ (𝐼 ∖ (𝐺 supp 0))) → (𝐺‘𝐴) = 0)
83	57, 72, 76, 82	syl21anc 835	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐺‘𝐴) = 0)
84	83, 70	eqeltrd 2837	. . . . 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 7347	. . . . . 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 12186	. . . . . 6 ⊢ (0 − 0) = 0
88	86, 87	eqtrdi 2792	. . . . 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 15094	. . . 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 2776	. . 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 24670	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → 𝐷 ∈ (Met‘𝑋))
92	91	ad3antrrr 727	. . . 4 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐷 ∈ (Met‘𝑋))
93		metge0 23596	. . . 4 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 0 ≤ (𝐹𝐷𝐺))
94	92, 58, 72, 93	syl3anc 1370	. . 3 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 0 ≤ (𝐹𝐷𝐺))
95	90, 94	eqbrtrd 5111	. 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 766	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐴 ∈ 𝐼)
97		simprl 768	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐹 ∈ 𝑋)
98	4, 97	rrxsuppss 24665	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹 supp 0) ⊆ 𝐼)
99		simprr 770	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐺 ∈ 𝑋)
100	4, 99	rrxsuppss 24665	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺 supp 0) ⊆ 𝐼)
101	98, 100	unssd 4132	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐼)
102		undif 4427	. . . . 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 217	. . . 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 2840	. . 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 4094	. . 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 217	. 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 956	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 396 ∨ wo 844 = wceq 1540 ∈ wcel 2105 {crab 3403 ∖ cdif 3894 ∪ cun 3895 ⊆ wss 3897 class class class wbr 5089 × cxp 5612 ↾ cres 5616 ∘ ccom 5618 ‘cfv 6473 (class class class)co 7329 supp csupp 8039 ↑_m cmap 8678 Fincfn 8796 finSupp cfsupp 9218 ℝcr 10963 0cc0 10964 ≤ cle 11103 − cmin 11298 2c2 12121 ↑cexp 13875 √csqrt 15035 abscabs 15036 Σcsu 15488 distcds 17060 Metcmet 20681 ℝ^crrx 24645

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-inf2 9490 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-pre-sup 11042 ax-addf 11043 ax-mulf 11044

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-se 5570 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-isom 6482 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-of 7587 df-om 7773 df-1st 7891 df-2nd 7892 df-supp 8040 df-tpos 8104 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-er 8561 df-map 8680 df-ixp 8749 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-fsupp 9219 df-sup 9291 df-oi 9359 df-card 9788 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-div 11726 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-7 12134 df-8 12135 df-9 12136 df-n0 12327 df-z 12413 df-dec 12531 df-uz 12676 df-rp 12824 df-xneg 12941 df-xadd 12942 df-xmul 12943 df-ico 13178 df-fz 13333 df-fzo 13476 df-seq 13815 df-exp 13876 df-hash 14138 df-cj 14901 df-re 14902 df-im 14903 df-sqrt 15037 df-abs 15038 df-clim 15288 df-sum 15489 df-struct 16937 df-sets 16954 df-slot 16972 df-ndx 16984 df-base 17002 df-ress 17031 df-plusg 17064 df-mulr 17065 df-starv 17066 df-sca 17067 df-vsca 17068 df-ip 17069 df-tset 17070 df-ple 17071 df-ds 17073 df-unif 17074 df-hom 17075 df-cco 17076 df-0g 17241 df-gsum 17242 df-prds 17247 df-pws 17249 df-mgm 18415 df-sgrp 18464 df-mnd 18475 df-mhm 18519 df-grp 18668 df-minusg 18669 df-sbg 18670 df-subg 18840 df-ghm 18920 df-cntz 19011 df-cmn 19475 df-abl 19476 df-mgp 19808 df-ur 19825 df-ring 19872 df-cring 19873 df-oppr 19949 df-dvdsr 19970 df-unit 19971 df-invr 20001 df-dvr 20012 df-rnghom 20046 df-drng 20087 df-field 20088 df-subrg 20119 df-staf 20203 df-srng 20204 df-lmod 20223 df-lss 20292 df-sra 20532 df-rgmod 20533 df-xmet 20688 df-met 20689 df-cnfld 20696 df-refld 20908 df-dsmm 21037 df-frlm 21052 df-nm 23836 df-tng 23838 df-tcph 24431 df-rrx 24647

This theorem is referenced by: rrnprjdstle 44167

W3C validator