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Theorem rrxdstprj1 23000

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 794	. . 3 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) → 𝐼 ∈ 𝑉)
2		simpr 476	. . 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 788	. . 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 790	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐹 ∈ 𝑋)
6	4, 5	rrxfsupp 22993	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹 supp 0) ∈ Fin)
7		simprr 792	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐺 ∈ 𝑋)
8	4, 7	rrxfsupp 22993	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺 supp 0) ∈ Fin)
9		unfi 8112	. . . . . . . 8 ⊢ (((𝐹 supp 0) ∈ Fin ∧ (𝐺 supp 0) ∈ Fin) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ∈ Fin)
10	6, 8, 9	syl2anc 691	. . . . . . 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 22994	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹 supp 0) ⊆ 𝐼)
12	4, 7	rrxsuppss 22994	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺 supp 0) ⊆ 𝐼)
13	11, 12	unssd 3751	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐼)
14	13	sselda 3568	. . . . . . . 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 22992	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐹:𝐼⟶ℝ)
16	15	ffvelrnda 6267	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ ℝ)
17	4, 7	rrxf 22992	. . . . . . . . . . 11 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐺:𝐼⟶ℝ)
18	17	ffvelrnda 6267	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → (𝐺‘𝑘) ∈ ℝ)
19	16, 18	resubcld 10337	. . . . . . . . 9 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝑘 ∈ 𝐼) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℝ)
20	19	resqcld 12897	. . . . . . . 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 486	. . . . . . 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 12898	. . . . . . . 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 486	. . . . . . 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 6103	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (𝐹‘𝑘) = (𝐹‘𝐴))
25		fveq2 6103	. . . . . . . . 9 ⊢ (𝑘 = 𝐴 → (𝐺‘𝑘) = (𝐺‘𝐴))
26	24, 25	oveq12d 6567	. . . . . . . 8 ⊢ (𝑘 = 𝐴 → ((𝐹‘𝑘) − (𝐺‘𝑘)) = ((𝐹‘𝐴) − (𝐺‘𝐴)))
27	26	oveq1d 6564	. . . . . . 7 ⊢ (𝑘 = 𝐴 → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) = (((𝐹‘𝐴) − (𝐺‘𝐴))↑2))
28		simplr 788	. . . . . . 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 14370	. . . . . 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 3569	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐴 ∈ 𝐼)
31	15, 30	ffvelrnd 6268	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹‘𝐴) ∈ ℝ)
32	17, 30	ffvelrnd 6268	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺‘𝐴) ∈ ℝ)
33	31, 32	resubcld 10337	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴) − (𝐺‘𝐴)) ∈ ℝ)
34		absresq 13890	. . . . . . 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 14312	. . . . . . 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 14368	. . . . . . 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 13844	. . . . . . 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 691	. . . . . 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 4615	. . . . 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 9947	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴) − (𝐺‘𝐴)) ∈ ℂ)
42	41	abscld 14023	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (abs‘((𝐹‘𝐴) − (𝐺‘𝐴))) ∈ ℝ)
43	36, 37	resqrtcld 14004	. . . . . 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 14031	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 0 ≤ (abs‘((𝐹‘𝐴) − (𝐺‘𝐴))))
45	36, 37	sqrtge0d 14007	. . . . . 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 12906	. . . . 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 246	. . . 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 22400	. . . . 5 ⊢ (((𝐹‘𝐴) ∈ ℝ ∧ (𝐺‘𝐴) ∈ ℝ) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) = (abs‘((𝐹‘𝐴) − (𝐺‘𝐴))))
50	31, 32, 49	syl2anc 691	. . . 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 22996	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
53	52	3expb 1258	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
54	53	adantlr 747	. . . 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 4615	. . 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 1317	. 2 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) ≤ (𝐹𝐷𝐺))
57		simplll 794	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐼 ∈ 𝑉)
58		simplrl 796	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐹 ∈ 𝑋)
59		ssun1 3738	. . . . . . . . . 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 3709	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ⊆ (𝐼 ∖ (𝐹 supp 0)))
62	61	sselda 3568	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐴 ∈ (𝐼 ∖ (𝐹 supp 0)))
63		simpr 476	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 𝐹 ∈ 𝑋)
64	4, 63	rrxf 22992	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 𝐹:𝐼⟶ℝ)
65		ssid 3587	. . . . . . . . 9 ⊢ (𝐹 supp 0) ⊆ (𝐹 supp 0)
66	65	a1i 11	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → (𝐹 supp 0) ⊆ (𝐹 supp 0))
67		simpl 472	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 𝐼 ∈ 𝑉)
68		0red 9920	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) → 0 ∈ ℝ)
69	64, 66, 67, 68	suppssr 7213	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋) ∧ 𝐴 ∈ (𝐼 ∖ (𝐹 supp 0))) → (𝐹‘𝐴) = 0)
70	57, 58, 62, 69	syl21anc 1317	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐹‘𝐴) = 0)
71		0red 9920	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 0 ∈ ℝ)
72	70, 71	eqeltrd 2688	. . . . 5 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐹‘𝐴) ∈ ℝ)
73		simplrr 797	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐺 ∈ 𝑋)
74		ssun2 3739	. . . . . . . . . 10 ⊢ (𝐺 supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0))
75	74	a1i 11	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺 supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0)))
76	75	sscond 3709	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0))) ⊆ (𝐼 ∖ (𝐺 supp 0)))
77	76	sselda 3568	. . . . . . 7 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐴 ∈ (𝐼 ∖ (𝐺 supp 0)))
78		simpr 476	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 𝐺 ∈ 𝑋)
79	4, 78	rrxf 22992	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 𝐺:𝐼⟶ℝ)
80		ssid 3587	. . . . . . . . 9 ⊢ (𝐺 supp 0) ⊆ (𝐺 supp 0)
81	80	a1i 11	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → (𝐺 supp 0) ⊆ (𝐺 supp 0))
82		simpl 472	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 𝐼 ∈ 𝑉)
83		0red 9920	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) → 0 ∈ ℝ)
84	79, 81, 82, 83	suppssr 7213	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐺 ∈ 𝑋) ∧ 𝐴 ∈ (𝐼 ∖ (𝐺 supp 0))) → (𝐺‘𝐴) = 0)
85	57, 73, 77, 84	syl21anc 1317	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐺‘𝐴) = 0)
86	85, 71	eqeltrd 2688	. . . . 5 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (𝐺‘𝐴) ∈ ℝ)
87	72, 86, 49	syl2anc 691	. . . 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))))
88	70, 85	oveq12d 6567	. . . . . 6 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → ((𝐹‘𝐴) − (𝐺‘𝐴)) = (0 − 0))
89		0m0e0 11007	. . . . . 6 ⊢ (0 − 0) = 0
90	88, 89	syl6eq 2660	. . . . 5 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → ((𝐹‘𝐴) − (𝐺‘𝐴)) = 0)
91	90	abs00bd 13879	. . . 4 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → (abs‘((𝐹‘𝐴) − (𝐺‘𝐴))) = 0)
92	87, 91	eqtrd 2644	. . 3 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) = 0)
93	4, 51	rrxmet 22999	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → 𝐷 ∈ (Met‘𝑋))
94	93	ad3antrrr 762	. . . 4 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 𝐷 ∈ (Met‘𝑋))
95		metge0 21960	. . . 4 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 0 ≤ (𝐹𝐷𝐺))
96	94, 58, 73, 95	syl3anc 1318	. . 3 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → 0 ≤ (𝐹𝐷𝐺))
97	92, 96	eqbrtrd 4605	. 2 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) ∧ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) ≤ (𝐹𝐷𝐺))
98		simplr 788	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐴 ∈ 𝐼)
99		simprl 790	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐹 ∈ 𝑋)
100	4, 99	rrxsuppss 22994	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐹 supp 0) ⊆ 𝐼)
101		simprr 792	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐺 ∈ 𝑋)
102	4, 101	rrxsuppss 22994	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐺 supp 0) ⊆ 𝐼)
103	100, 102	unssd 3751	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐼)
104		undif 4001	. . . . 5 ⊢ (((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐼 ↔ (((𝐹 supp 0) ∪ (𝐺 supp 0)) ∪ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) = 𝐼)
105	103, 104	sylib 207	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (((𝐹 supp 0) ∪ (𝐺 supp 0)) ∪ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))) = 𝐼)
106	98, 105	eleqtrrd 2691	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → 𝐴 ∈ (((𝐹 supp 0) ∪ (𝐺 supp 0)) ∪ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))))
107		elun 3715	. . 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)))))
108	106, 107	sylib 207	. 2 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → (𝐴 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0)) ∨ 𝐴 ∈ (𝐼 ∖ ((𝐹 supp 0) ∪ (𝐺 supp 0)))))
109	56, 97, 108	mpjaodan 823	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 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 class class class wbr 4583 × cxp 5036 ↾ cres 5040 ∘ ccom 5042 ‘cfv 5804 (class class class)co 6549 supp csupp 7182 ↑_𝑚 cmap 7744 Fincfn 7841 finSupp cfsupp 8158 ℝcr 9814 0cc0 9815 ≤ cle 9954 − cmin 10145 2c2 10947 ↑cexp 12722 √csqrt 13821 abscabs 13822 Σcsu 14264 distcds 15777 Metcme 19553 ℝ^crrx 22979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ico 12052 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-0g 15925 df-gsum 15926 df-prds 15931 df-pws 15933 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-ghm 17481 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-rnghom 18538 df-drng 18572 df-field 18573 df-subrg 18601 df-staf 18668 df-srng 18669 df-lmod 18688 df-lss 18754 df-sra 18993 df-rgmod 18994 df-xmet 19560 df-met 19561 df-cnfld 19568 df-refld 19770 df-dsmm 19895 df-frlm 19910 df-nm 22197 df-tng 22199 df-tch 22777 df-rrx 22981

This theorem is referenced by: rrnprjdstle 39197

W3C validator