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Theorem rrxmval 25452

Description: The value of the Euclidean metric. Compare with rrnmval 37814. (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 2734	. . . . . 6 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
3		eqid 2734	. . . . . 6 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
4	2, 3	rrxds 25440	. . . . 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 2793	. . . 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 25435	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (Base‘(ℝ^‘𝐼)) = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0})
8	6, 7	eqtr4id 2793	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → 𝑋 = (Base‘(ℝ^‘𝐼)))
9		mpoeq12 7505	. . . . 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 584	. . . 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 2777	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))))
12	11	3ad2ant1 1132	. 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 771	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → 𝑓 = 𝐹)
14	13	fveq1d 6908	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑓‘𝑥) = (𝐹‘𝑥))
15		simprr 773	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → 𝑔 = 𝐺)
16	15	fveq1d 6908	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑔‘𝑥) = (𝐺‘𝑥))
17	14, 16	oveq12d 7448	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → ((𝑓‘𝑥) − (𝑔‘𝑥)) = ((𝐹‘𝑥) − (𝐺‘𝑥)))
18	17	oveq1d 7445	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (((𝑓‘𝑥) − (𝑔‘𝑥))↑2) = (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))
19	18	mpteq2dv 5249	. . . . 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 7446	. . . 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 1136	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐹 ∈ 𝑋)
22	6, 21	rrxf 25448	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐹:𝐼⟶ℝ)
23	22	ffvelcdmda 7103	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℝ)
24		simp3 1137	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐺 ∈ 𝑋)
25	6, 24	rrxf 25448	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐺:𝐼⟶ℝ)
26	25	ffvelcdmda 7103	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ ℝ)
27	23, 26	resubcld 11688	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥) − (𝐺‘𝑥)) ∈ ℝ)
28	27	resqcld 14161	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → (((𝐹‘𝑥) − (𝐺‘𝑥))↑2) ∈ ℝ)
29	28	fmpttd 7134	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)):𝐼⟶ℝ)
30	6, 21	rrxfsupp 25449	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹 supp 0) ∈ Fin)
31	6, 24	rrxfsupp 25449	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐺 supp 0) ∈ Fin)
32		unfi 9209	. . . . . . . . . 10 ⊢ (((𝐹 supp 0) ∈ Fin ∧ (𝐺 supp 0) ∈ Fin) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ∈ Fin)
33	30, 31, 32	syl2anc 584	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ∈ Fin)
34	6	rrxmvallem 25451	. . . . . . . . 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 9298	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0) ∈ Fin)
36		mptexg 7240	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝑉 → (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) ∈ V)
37		funmpt 6605	. . . . . . . . . . 11 ⊢ Fun (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))
38		0cn 11250	. . . . . . . . . . 11 ⊢ 0 ∈ ℂ
39		funisfsupp 9404	. . . . . . . . . . 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 1451	. . . . . . . . . 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 17	. . . . . . . . 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 1132	. . . . . . . 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 257	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) finSupp 0)
44		simp1 1135	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐼 ∈ 𝑉)
45		regsumsupp 21657	. . . . . . 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 1370	. . . . . 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 8200	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0) ⊆ dom (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))
48		eqid 2734	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) = (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))
49	48	dmmptss 6262	. . . . . . . . . . 11 ⊢ dom (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) ⊆ 𝐼
50	47, 49	sstri 4004	. . . . . . . . . 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 3994	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0)) → 𝑘 ∈ 𝐼)
53		eqidd 2735	. . . . . . . . . 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 484	. . . . . . . . . . . . 13 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → 𝑥 = 𝑘)
55	54	fveq2d 6910	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → (𝐹‘𝑥) = (𝐹‘𝑘))
56	54	fveq2d 6910	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → (𝐺‘𝑥) = (𝐺‘𝑘))
57	55, 56	oveq12d 7448	. . . . . . . . . . 11 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → ((𝐹‘𝑥) − (𝐺‘𝑥)) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
58	57	oveq1d 7445	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → (((𝐹‘𝑥) − (𝐺‘𝑥))↑2) = (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
59		simpr 484	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
60		ovexd 7465	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) ∈ V)
61	53, 58, 59, 60	fvmptd 7022	. . . . . . . . 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 2740	. . . . . . . 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 591	. . . . . . 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 15734	. . . . . 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 2777	. . . . 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 480	. . . 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 7103	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ ℝ)
68	67	recnd 11286	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ ℂ)
69	25	ffvelcdmda 7103	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝐺‘𝑘) ∈ ℝ)
70	69	recnd 11286	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝐺‘𝑘) ∈ ℂ)
71	68, 70	subcld 11617	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℂ)
72	71	sqcld 14180	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) ∈ ℂ)
73	52, 72	syldan 591	. . . . . 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 25450	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹 supp 0) ⊆ 𝐼)
75	6, 24	rrxsuppss 25450	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐺 supp 0) ⊆ 𝐼)
76	74, 75	unssd 4201	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐼)
77	76	ssdifssd 4156	. . . . . . . . 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 3994	. . . . . . . 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 591	. . . . . . 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 4154	. . . . . . . . 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 3994	. . . . . . . 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 4018	. . . . . . . . 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 11251	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 0 ∈ ℂ)
84	29, 82, 44, 83	suppssr 8218	. . . . . . . 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 591	. . . . . . 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 2774	. . . . . 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 15757	. . . . 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 480	. . . 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 2778	. . 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 6910	. 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 6921	. 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 7584	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

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 {crab 3432 Vcvv 3477 ∖ cdif 3959 ∪ cun 3960 ⊆ wss 3962 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5688 Fun wfun 6556 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ∈ cmpo 7432 supp csupp 8183 ↑_m cmap 8864 Fincfn 8983 finSupp cfsupp 9398 ℂcc 11150 ℝcr 11151 0cc0 11152 − cmin 11489 2c2 12318 ↑cexp 14098 √csqrt 15268 Σcsu 15718 Basecbs 17244 distcds 17306 Σ_g cgsu 17486 ℝ_fldcrefld 21639 ℝ^crrx 25430

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 ax-mulf 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-rp 13032 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 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 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-grp 18966 df-minusg 18967 df-sbg 18968 df-subg 19153 df-ghm 19243 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-dvr 20417 df-rhm 20488 df-subrng 20562 df-subrg 20586 df-drng 20747 df-field 20748 df-staf 20856 df-srng 20857 df-lmod 20876 df-lss 20947 df-sra 21189 df-rgmod 21190 df-cnfld 21382 df-refld 21640 df-dsmm 21769 df-frlm 21784 df-nm 24610 df-tng 24612 df-tcph 25216 df-rrx 25432

This theorem is referenced by: rrxmfval 25453 rrxmet 25455 rrxdstprj1 25456

W3C validator