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Theorem rrxmval 25615

Description: The value of the Euclidean metric. Compare with rrnmval 38537. (Contributed by Thierry Arnoux, 30-Jun-2019.)

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

**Assertion**
Ref	Expression
rrxmval	⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))

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

Proof of Theorem rrxmval Dummy variables 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrxmval.d	. . . . 5 ⊢ 𝐷 = (dist‘(ℝ^‘𝐼))
2		eqid 2765	. . . . . 6 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
3		eqid 2765	. . . . . 6 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
4	2, 3	rrxds 25603	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) = (dist‘(ℝ^‘𝐼)))
5	1, 4	eqtr4id 2819	. . . 4 ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))))
6		rrxmval.1	. . . . . 6 ⊢ 𝑋 = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0}
7	2, 3	rrxbase 25598	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (Base‘(ℝ^‘𝐼)) = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0})
8	6, 7	eqtr4id 2819	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → 𝑋 = (Base‘(ℝ^‘𝐼)))
9		mpoeq12 7492	. . . . 5 ⊢ ((𝑋 = (Base‘(ℝ^‘𝐼)) ∧ 𝑋 = (Base‘(ℝ^‘𝐼))) → (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) = (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))))
10	8, 8, 9	syl2anc 596	. . . 4 ⊢ (𝐼 ∈ 𝑉 → (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))) = (𝑓 ∈ (Base‘(ℝ^‘𝐼)), 𝑔 ∈ (Base‘(ℝ^‘𝐼)) ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))))
11	5, 10	eqtr4d 2803	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))))
12	11	3ad2ant1 1151	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))))
13		simprl 783	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → 𝑓 = 𝐹)
14	13	fveq1d 6887	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑓‘𝑥) = (𝐹‘𝑥))
15		simprr 785	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → 𝑔 = 𝐺)
16	15	fveq1d 6887	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑔‘𝑥) = (𝐺‘𝑥))
17	14, 16	oveq12d 7437	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → ((𝑓‘𝑥) − (𝑔‘𝑥)) = ((𝐹‘𝑥) − (𝐺‘𝑥)))
18	17	oveq1d 7434	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (((𝑓‘𝑥) − (𝑔‘𝑥))↑2) = (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))
19	18	mpteq2dv 5207	. . . . 5 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)) = (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)))
20	19	oveq2d 7435	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))) = (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))))
21		simp2 1155	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐹 ∈ 𝑋)
22	6, 21	rrxf 25611	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐹:𝐼⟶ℝ)
23	22	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℝ)
24		simp3 1156	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐺 ∈ 𝑋)
25	6, 24	rrxf 25611	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐺:𝐼⟶ℝ)
26	25	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ ℝ)
27	23, 26	resubcld 11657	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥) − (𝐺‘𝑥)) ∈ ℝ)
28	27	resqcld 14179	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → (((𝐹‘𝑥) − (𝐺‘𝑥))↑2) ∈ ℝ)
29	28	fmpttd 7114	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)):𝐼⟶ℝ)
30	6, 21	rrxfsupp 25612	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹 supp 0) ∈ Fin)
31	6, 24	rrxfsupp 25612	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐺 supp 0) ∈ Fin)
32		unfi 9162	. . . . . . . . . 10 ⊢ (((𝐹 supp 0) ∈ Fin ∧ (𝐺 supp 0) ∈ Fin) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ∈ Fin)
33	30, 31, 32	syl2anc 596	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ∈ Fin)
34	6	rrxmvallem 25614	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0) ⊆ ((𝐹 supp 0) ∪ (𝐺 supp 0)))
35	33, 34	ssfid 9236	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0) ∈ Fin)
36		mptexg 7226	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) ∈ V)
37		funmpt 6578	. . . . . . . . . . 11 ⊢ Fun (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))
38		0cn 11213	. . . . . . . . . . 11 ⊢ 0 ∈ ℂ
39		funisfsupp 9334	. . . . . . . . . . 11 ⊢ ((Fun (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) ∧ (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) ∈ V ∧ 0 ∈ ℂ) → ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) finSupp 0 ↔ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0) ∈ Fin))
40	37, 38, 39	mp3an13 1481	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) ∈ V → ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) finSupp 0 ↔ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0) ∈ Fin))
41	36, 40	syl 18	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝑉 → ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) finSupp 0 ↔ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0) ∈ Fin))
42	41	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) finSupp 0 ↔ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0) ∈ Fin))
43	35, 42	mpbird 260	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) finSupp 0)
44		simp1 1154	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐼 ∈ 𝑉)
45		regsumsupp 21822	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)):𝐼⟶ℝ ∧ (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) finSupp 0 ∧ 𝐼 ∈ 𝑉) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))) = Σ𝑘 ∈ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0)((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))‘𝑘))
46	29, 43, 44, 45	syl3anc 1398	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))) = Σ𝑘 ∈ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0)((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))‘𝑘))
47		suppssdm 8179	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0) ⊆ dom (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))
48		eqid 2765	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) = (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))
49	48	dmmptss 6244	. . . . . . . . . . 11 ⊢ dom (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) ⊆ 𝐼
50	47, 49	sstri 3947	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0) ⊆ 𝐼
51	50	a1i 11	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0) ⊆ 𝐼)
52	51	sselda 3938	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0)) → 𝑘 ∈ 𝐼)
53		eqidd 2766	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) = (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)))
54		simpr 490	. . . . . . . . . . . . 13 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → 𝑥 = 𝑘)
55	54	fveq2d 6889	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → (𝐹‘𝑥) = (𝐹‘𝑘))
56	54	fveq2d 6889	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → (𝐺‘𝑥) = (𝐺‘𝑘))
57	55, 56	oveq12d 7437	. . . . . . . . . . 11 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → ((𝐹‘𝑥) − (𝐺‘𝑥)) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
58	57	oveq1d 7434	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → (((𝐹‘𝑥) − (𝐺‘𝑥))↑2) = (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
59		simpr 490	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
60		ovexd 7454	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) ∈ V)
61	53, 58, 59, 60	fvmptd 7001	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))‘𝑘) = (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
62	61	eqcomd 2771	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) = ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))‘𝑘))
63	52, 62	syldan 603	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0)) → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) = ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))‘𝑘))
64	63	sumeq2dv 15777	. . . . . 6 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → Σ𝑘 ∈ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0)(((𝐹‘𝑘) − (𝐺‘𝑘))↑2) = Σ𝑘 ∈ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0)((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))‘𝑘))
65	46, 64	eqtr4d 2803	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))) = Σ𝑘 ∈ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0)(((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
66	65	adantr 486	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))) = Σ𝑘 ∈ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0)(((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
67	22	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ ℝ)
68	67	recnd 11252	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ ℂ)
69	25	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝐺‘𝑘) ∈ ℝ)
70	69	recnd 11252	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝐺‘𝑘) ∈ ℂ)
71	68, 70	subcld 11584	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℂ)
72	71	sqcld 14198	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) ∈ ℂ)
73	52, 72	syldan 603	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0)) → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) ∈ ℂ)
74	6, 21	rrxsuppss 25613	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹 supp 0) ⊆ 𝐼)
75	6, 24	rrxsuppss 25613	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐺 supp 0) ⊆ 𝐼)
76	74, 75	unssd 4145	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐼)
77	76	ssdifssd 4101	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (((𝐹 supp 0) ∪ (𝐺 supp 0)) ∖ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0)) ⊆ 𝐼)
78	77	sselda 3938	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ (((𝐹 supp 0) ∪ (𝐺 supp 0)) ∖ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0))) → 𝑘 ∈ 𝐼)
79	78, 62	syldan 603	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ (((𝐹 supp 0) ∪ (𝐺 supp 0)) ∖ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0))) → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) = ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))‘𝑘))
80	76	ssdifd 4099	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (((𝐹 supp 0) ∪ (𝐺 supp 0)) ∖ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0)) ⊆ (𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0)))
81	80	sselda 3938	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ (((𝐹 supp 0) ∪ (𝐺 supp 0)) ∖ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0))) → 𝑘 ∈ (𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0)))
82		ssidd 3961	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0) ⊆ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0))
83		0cnd 11214	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 0 ∈ ℂ)
84	29, 82, 44, 83	suppssr 8197	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ (𝐼 ∖ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0))) → ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))‘𝑘) = 0)
85	81, 84	syldan 603	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ (((𝐹 supp 0) ∪ (𝐺 supp 0)) ∖ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0))) → ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))‘𝑘) = 0)
86	79, 85	eqtrd 2800	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ (((𝐹 supp 0) ∪ (𝐺 supp 0)) ∖ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0))) → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) = 0)
87	34, 73, 86, 33	fsumss 15799	. . . . 5 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → Σ𝑘 ∈ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0)(((𝐹‘𝑘) − (𝐺‘𝑘))↑2) = Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
88	87	adantr 486	. . . 4 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → Σ𝑘 ∈ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0)(((𝐹‘𝑘) − (𝐺‘𝑘))↑2) = Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
89	20, 66, 88	3eqtrd 2804	. . 3 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))) = Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
90	89	fveq2d 6889	. 2 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2)))) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))
91		fvexd 6900	. 2 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)) ∈ V)
92	12, 90, 21, 24, 91	ovmpod 7571	1 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹𝐷𝐺) = (√‘Σ𝑘 ∈ ((𝐹 supp 0) ∪ (𝐺 supp 0))(((𝐹‘𝑘) − (𝐺‘𝑘))↑2)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 {crab 3418 Vcvv 3457 ∖ cdif 3903 ∪ cun 3904 ⊆ wss 3906 class class class wbr 5111 ↦ cmpt 5194 dom cdm 5663 Fun wfun 6534 ⟶wf 6536 ‘cfv 6540 (class class class)co 7419 ∈ cmpo 7421 supp csupp 8162 ↑_m cmap 8830 Fincfn 8949 finSupp cfsupp 9328 ℂcc 11113 ℝcr 11114 0cc0 11115 − cmin 11456 2c2 12310 ↑cexp 14115 √csqrt 15308 Σcsu 15761 Basecbs 17291 distcds 17341 Σ_g cgsu 17515 ℝ_fldcrefld 21804 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-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-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: rrxmfval 25616 rrxmet 25618 rrxdstprj1 25619

W3C validator