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Theorem rrxdstprj1 25443

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 775	. . 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 769	. . 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 771	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐹 ∈ 𝑋)
6	4, 5	rrxfsupp 25436	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹 supp 0) ∈ Fin)
7		simprr 773	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐺 ∈ 𝑋)
8	4, 7	rrxfsupp 25436	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺 supp 0) ∈ Fin)
9		unfi 9211	. . . . . . . 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 25437	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹 supp 0) ⊆ 𝐼)
12	4, 7	rrxsuppss 25437	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺 supp 0) ⊆ 𝐼)
13	11, 12	unssd 4192	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐼)
14	13	sselda 3983	. . . . . . . 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 25435	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐹:𝐼⟶ℝ)
16	15	ffvelcdmda 7104	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ ℝ)
17	4, 7	rrxf 25435	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐺:𝐼⟶ℝ)
18	17	ffvelcdmda 7104	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝐺‘𝑘) ∈ ℝ)
19	16, 18	resubcld 11691	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℝ)
20	19	resqcld 14165	. . . . . . . 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 14177	. . . . . . . 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 6906	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (𝐹‘𝑘) = (𝐹‘𝐴))
25		fveq2 6906	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (𝐺‘𝑘) = (𝐺‘𝐴))
26	24, 25	oveq12d 7449	. . . . . . . 8 ⊢ (𝑘 = 𝐴 → ((𝐹‘𝑘) − (𝐺‘𝑘)) = ((𝐹‘𝐴) − (𝐺‘𝐴)))
27	26	oveq1d 7446	. . . . . . 7 ⊢ (𝑘 = 𝐴 → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) = (((𝐹‘𝐴) − (𝐺‘𝐴))↑2))
28		simplr 769	. . . . . . 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 15833	. . . . . 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 3984	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐴 ∈ 𝐼)
31	15, 30	ffvelcdmd 7105	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹‘𝐴) ∈ ℝ)
32	17, 30	ffvelcdmd 7105	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺‘𝐴) ∈ ℝ)
33	31, 32	resubcld 11691	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴) − (𝐺‘𝐴)) ∈ ℝ)
34		absresq 15341	. . . . . . 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 15770	. . . . . . 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 15831	. . . . . . 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 15294	. . . . . . 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 5175	. . . . 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 11289	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴) − (𝐺‘𝐴)) ∈ ℂ)
42	41	abscld 15475	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (abs‘((𝐹‘𝐴) − (𝐺‘𝐴))) ∈ ℝ)
43	36, 37	resqrtcld 15456	. . . . . 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 15483	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 0 ≤ (abs‘((𝐹‘𝐴) − (𝐺‘𝐴))))
45	36, 37	sqrtge0d 15459	. . . . . 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 14296	. . . . 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 24810	. . . . 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 25439	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
53	52	3expb 1121	. . . . 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 5175	. . 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 838	. 2 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) ≤ (𝐹𝐷𝐺))
57		simplll 775	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐼 ∈ 𝑉)
58		simplrl 777	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐹 ∈ 𝑋)
59		ssun1 4178	. . . . . . . . . 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 4146	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ⊆ (𝐼 ∖ (𝐹 supp 0)))
62	61	sselda 3983	. . . . . . 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 25435	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 𝐹:𝐼⟶ℝ)
65		ssidd 4007	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → (𝐹 supp 0) ⊆ (𝐹 supp 0))
66		simpl 482	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 𝐼 ∈ 𝑉)
67		0red 11264	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 0 ∈ ℝ)
68	64, 65, 66, 67	suppssr 8220	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) ∧ 𝐴 ∈ (𝐼 ∖ (𝐹 supp 0))) → (𝐹‘𝐴) = 0)
69	57, 58, 62, 68	syl21anc 838	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐹‘𝐴) = 0)
70		0red 11264	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 0 ∈ ℝ)
71	69, 70	eqeltrd 2841	. . . . 5 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐹‘𝐴) ∈ ℝ)
72		simplrr 778	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐺 ∈ 𝑋)
73		ssun2 4179	. . . . . . . . . 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 4146	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ⊆ (𝐼 ∖ (𝐺 supp 0)))
76	75	sselda 3983	. . . . . . 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 25435	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 𝐺:𝐼⟶ℝ)
79		ssidd 4007	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → (𝐺 supp 0) ⊆ (𝐺 supp 0))
80		simpl 482	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 𝐼 ∈ 𝑉)
81		0red 11264	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 0 ∈ ℝ)
82	78, 79, 80, 81	suppssr 8220	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) ∧ 𝐴 ∈ (𝐼 ∖ (𝐺 supp 0))) → (𝐺‘𝐴) = 0)
83	57, 72, 76, 82	syl21anc 838	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐺‘𝐴) = 0)
84	83, 70	eqeltrd 2841	. . . . 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 7449	. . . . . 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 12386	. . . . . 6 ⊢ (0 − 0) = 0
88	86, 87	eqtrdi 2793	. . . . 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 15330	. . . 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 2777	. . 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 25442	. . . . 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 24355	. . . 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 5165	. 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 769	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐴 ∈ 𝐼)
97		simprl 771	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐹 ∈ 𝑋)
98	4, 97	rrxsuppss 25437	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹 supp 0) ⊆ 𝐼)
99		simprr 773	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐺 ∈ 𝑋)
100	4, 99	rrxsuppss 25437	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺 supp 0) ⊆ 𝐼)
101	98, 100	unssd 4192	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐼)
102		undif 4482	. . . . 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 2844	. . 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 4153	. . 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 961	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 848 = wceq 1540 ∈ wcel 2108 {crab 3436 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 class class class wbr 5143 × cxp 5683 ↾ cres 5687 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 supp csupp 8185 ↑_m cmap 8866 Fincfn 8985 finSupp cfsupp 9401 ℝcr 11154 0cc0 11155 ≤ cle 11296 − cmin 11492 2c2 12321 ↑cexp 14102 √csqrt 15272 abscabs 15273 Σcsu 15722 distcds 17306 Metcmet 21350 ℝ^crrx 25417

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ico 13393 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-0g 17486 df-gsum 17487 df-prds 17492 df-pws 17494 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-ghm 19231 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-subrng 20546 df-subrg 20570 df-drng 20731 df-field 20732 df-staf 20840 df-srng 20841 df-lmod 20860 df-lss 20930 df-sra 21172 df-rgmod 21173 df-xmet 21357 df-met 21358 df-cnfld 21365 df-refld 21623 df-dsmm 21752 df-frlm 21767 df-nm 24595 df-tng 24597 df-tcph 25203 df-rrx 25419

This theorem is referenced by: rrnprjdstle 46316

W3C validator