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Theorem rrxmval 25719

Description: The value of the Euclidean metric. Compare with rrnmval 38742. (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 2761	. . . . . 6 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
3		eqid 2761	. . . . . 6 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
4	2, 3	rrxds 25707	. . . . 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 2815	. . . 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 25702	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (Base‘(ℝ^‘𝐼)) = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0})
8	6, 7	eqtr4id 2815	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → 𝑋 = (Base‘(ℝ^‘𝐼)))
9		mpoeq12 7491	. . . . 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 2799	. . 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 6885	. . . . . . . 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 6885	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑔‘𝑥) = (𝐺‘𝑥))
17	14, 16	oveq12d 7436	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → ((𝑓‘𝑥) − (𝑔‘𝑥)) = ((𝐹‘𝑥) − (𝐺‘𝑥)))
18	17	oveq1d 7433	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (((𝑓‘𝑥) − (𝑔‘𝑥))↑2) = (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))
19	18	mpteq2dv 5199	. . . . 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 7434	. . . 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 25715	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐹:𝐼⟶ℝ)
23	22	ffvelcdmda 7082	. . . . . . . . . 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 25715	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐺:𝐼⟶ℝ)
26	25	ffvelcdmda 7082	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ ℝ)
27	23, 26	resubcld 11737	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥) − (𝐺‘𝑥)) ∈ ℝ)
28	27	resqcld 14261	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → (((𝐹‘𝑥) − (𝐺‘𝑥))↑2) ∈ ℝ)
29	28	fmpttd 7113	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)):𝐼⟶ℝ)
30	6, 21	rrxfsupp 25716	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹 supp 0) ∈ Fin)
31	6, 24	rrxfsupp 25716	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐺 supp 0) ∈ Fin)
32		unfi 9179	. . . . . . . . . 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 25718	. . . . . . . . 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 9253	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0) ∈ Fin)
36		mptexg 7225	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) ∈ V)
37		funmpt 6576	. . . . . . . . . . 11 ⊢ Fun (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))
38		0cn 11291	. . . . . . . . . . 11 ⊢ 0 ∈ ℂ
39		funisfsupp 9352	. . . . . . . . . . 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 21921	. . . . . . 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 8187	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0) ⊆ dom (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))
48		eqid 2761	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) = (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))
49	48	dmmptss 6241	. . . . . . . . . . 11 ⊢ dom (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) ⊆ 𝐼
50	47, 49	sstri 3940	. . . . . . . . . 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 3931	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0)) → 𝑘 ∈ 𝐼)
53		eqidd 2762	. . . . . . . . . 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 6887	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → (𝐹‘𝑥) = (𝐹‘𝑘))
56	54	fveq2d 6887	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → (𝐺‘𝑥) = (𝐺‘𝑘))
57	55, 56	oveq12d 7436	. . . . . . . . . . 11 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → ((𝐹‘𝑥) − (𝐺‘𝑥)) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
58	57	oveq1d 7433	. . . . . . . . . 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 7453	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) ∈ V)
61	53, 58, 59, 60	fvmptd 6999	. . . . . . . . 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 2767	. . . . . . . 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 15862	. . . . . 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 2799	. . . . 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 7082	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ ℝ)
68	67	recnd 11330	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ ℂ)
69	25	ffvelcdmda 7082	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝐺‘𝑘) ∈ ℝ)
70	69	recnd 11330	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝐺‘𝑘) ∈ ℂ)
71	68, 70	subcld 11662	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℂ)
72	71	sqcld 14280	. . . . . . 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 25717	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹 supp 0) ⊆ 𝐼)
75	6, 24	rrxsuppss 25717	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐺 supp 0) ⊆ 𝐼)
76	74, 75	unssd 4138	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐼)
77	76	ssdifssd 4094	. . . . . . . . 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 3931	. . . . . . . 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 4092	. . . . . . . . 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 3931	. . . . . . . 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 3954	. . . . . . . . 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 11292	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 0 ∈ ℂ)
84	29, 82, 44, 83	suppssr 8205	. . . . . . . 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 2796	. . . . . 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 15884	. . . . 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 2800	. . 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 6887	. 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 6898	. 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 7570	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 2145 {crab 3413 Vcvv 3451 ∖ cdif 3896 ∪ cun 3897 ⊆ wss 3899 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5651 Fun wfun 6531 ⟶wf 6533 ‘cfv 6537 (class class class)co 7418 ∈ cmpo 7420 supp csupp 8170 ↑_m cmap 8840 Fincfn 8966 finSupp cfsupp 9346 ℂcc 11191 ℝcr 11192 0cc0 11193 − cmin 11534 2c2 12390 ↑cexp 14197 √csqrt 15393 Σcsu 15846 Basecbs 17380 distcds 17430 Σ_g cgsu 17604 ℝ_fldcrefld 21903 ℝ^crrx 25697

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-inf2 9635 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271 ax-addf 11272 ax-mulf 11273

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-om 7876 df-1st 7999 df-2nd 8000 df-supp 8171 df-tpos 8236 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-er 8710 df-map 8842 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9347 df-sup 9427 df-oi 9497 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-z 12687 df-dec 12808 df-uz 12959 df-rp 13114 df-fz 13633 df-fzo 13782 df-seq 14138 df-exp 14198 df-hash 14468 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-clim 15648 df-sum 15847 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-starv 17436 df-sca 17437 df-vsca 17438 df-ip 17439 df-tset 17440 df-ple 17441 df-ds 17443 df-unif 17444 df-hom 17445 df-cco 17446 df-0g 17605 df-gsum 17606 df-prds 17611 df-pws 17613 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-mhm 18971 df-grp 19140 df-minusg 19141 df-sbg 19142 df-subg 19326 df-ghm 19421 df-cntz 19524 df-cmn 19989 df-abl 19990 df-mgp 20354 df-rng 20368 df-ur 20401 df-ring 20454 df-cring 20455 df-oppr 20560 df-dvdsr 20580 df-unit 20581 df-invr 20611 df-dvr 20624 df-rhm 20695 df-subrng 20791 df-subrg 20815 df-drng 20975 df-field 20976 df-staf 21089 df-srng 21090 df-lmod 21130 df-lss 21200 df-sra 21441 df-rgmod 21442 df-cnfld 21672 df-refld 21904 df-dsmm 22031 df-frlm 22046 df-nm 24894 df-tng 24896 df-tcph 25483 df-rrx 25699

This theorem is used by: rrxmfval 25720 rrxmet 25722 rrxdstprj1 25723

W3C validator