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Theorem rrxmval 25146

Description: The value of the Euclidean metric. Compare with rrnmval 36999. (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 2732	. . . . . 6 ⊢ (ℝ^‘𝐼) = (ℝ^‘𝐼)
3		eqid 2732	. . . . . 6 ⊢ (Base‘(ℝ^‘𝐼)) = (Base‘(ℝ^‘𝐼))
4	2, 3	rrxds 25134	. . . . 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 2791	. . . 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 25129	. . . . . 6 ⊢ (𝐼 ∈ 𝑉 → (Base‘(ℝ^‘𝐼)) = {ℎ ∈ (ℝ ↑_m 𝐼) ∣ ℎ finSupp 0})
8	6, 7	eqtr4id 2791	. . . . 5 ⊢ (𝐼 ∈ 𝑉 → 𝑋 = (Base‘(ℝ^‘𝐼)))
9		mpoeq12 7484	. . . . 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 2775	. . 3 ⊢ (𝐼 ∈ 𝑉 → 𝐷 = (𝑓 ∈ 𝑋, 𝑔 ∈ 𝑋 ↦ (√‘(ℝ_fld Σ_g (𝑥 ∈ 𝐼 ↦ (((𝑓‘𝑥) − (𝑔‘𝑥))↑2))))))
12	11	3ad2ant1 1133	. 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 769	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → 𝑓 = 𝐹)
14	13	fveq1d 6893	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑓‘𝑥) = (𝐹‘𝑥))
15		simprr 771	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → 𝑔 = 𝐺)
16	15	fveq1d 6893	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑔‘𝑥) = (𝐺‘𝑥))
17	14, 16	oveq12d 7429	. . . . . . 7 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → ((𝑓‘𝑥) − (𝑔‘𝑥)) = ((𝐹‘𝑥) − (𝐺‘𝑥)))
18	17	oveq1d 7426	. . . . . 6 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (((𝑓‘𝑥) − (𝑔‘𝑥))↑2) = (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))
19	18	mpteq2dv 5250	. . . . 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 7427	. . . 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 1137	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐹 ∈ 𝑋)
22	6, 21	rrxf 25142	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐹:𝐼⟶ℝ)
23	22	ffvelcdmda 7086	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → (𝐹‘𝑥) ∈ ℝ)
24		simp3 1138	. . . . . . . . . . . 12 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐺 ∈ 𝑋)
25	6, 24	rrxf 25142	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐺:𝐼⟶ℝ)
26	25	ffvelcdmda 7086	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → (𝐺‘𝑥) ∈ ℝ)
27	23, 26	resubcld 11646	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → ((𝐹‘𝑥) − (𝐺‘𝑥)) ∈ ℝ)
28	27	resqcld 14094	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑥 ∈ 𝐼) → (((𝐹‘𝑥) − (𝐺‘𝑥))↑2) ∈ ℝ)
29	28	fmpttd 7116	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)):𝐼⟶ℝ)
30	6, 21	rrxfsupp 25143	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹 supp 0) ∈ Fin)
31	6, 24	rrxfsupp 25143	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐺 supp 0) ∈ Fin)
32		unfi 9174	. . . . . . . . . 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 25145	. . . . . . . . 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 9269	. . . . . . . 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 6586	. . . . . . . . . . 11 ⊢ Fun (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))
38		0cn 11210	. . . . . . . . . . 11 ⊢ 0 ∈ ℂ
39		funisfsupp 9369	. . . . . . . . . . 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 1452	. . . . . . . . . 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 1133	. . . . . . . 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 256	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) finSupp 0)
44		simp1 1136	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 𝐼 ∈ 𝑉)
45		regsumsupp 21394	. . . . . . 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 1371	. . . . . 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 8164	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0) ⊆ dom (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))
48		eqid 2732	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) = (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2))
49	48	dmmptss 6240	. . . . . . . . . . 11 ⊢ dom (𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) ⊆ 𝐼
50	47, 49	sstri 3991	. . . . . . . . . 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 3982	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ ((𝑥 ∈ 𝐼 ↦ (((𝐹‘𝑥) − (𝐺‘𝑥))↑2)) supp 0)) → 𝑘 ∈ 𝐼)
53		eqidd 2733	. . . . . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → 𝑥 = 𝑘)
55	54	fveq2d 6895	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → (𝐹‘𝑥) = (𝐹‘𝑘))
56	54	fveq2d 6895	. . . . . . . . . . . 12 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → (𝐺‘𝑥) = (𝐺‘𝑘))
57	55, 56	oveq12d 7429	. . . . . . . . . . 11 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → ((𝐹‘𝑥) − (𝐺‘𝑥)) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
58	57	oveq1d 7426	. . . . . . . . . 10 ⊢ ((((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 = 𝑘) → (((𝐹‘𝑥) − (𝐺‘𝑥))↑2) = (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))
59		simpr 485	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝑘 ∈ 𝐼)
60		ovexd 7446	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (((𝐹‘𝑘) − (𝐺‘𝑘))↑2) ∈ V)
61	53, 58, 59, 60	fvmptd 7005	. . . . . . . . 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 2738	. . . . . . . 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 15653	. . . . . 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 2775	. . . . 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 481	. . . 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 7086	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ ℝ)
68	67	recnd 11246	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ ℂ)
69	25	ffvelcdmda 7086	. . . . . . . . . 10 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝐺‘𝑘) ∈ ℝ)
70	69	recnd 11246	. . . . . . . . 9 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → (𝐺‘𝑘) ∈ ℂ)
71	68, 70	subcld 11575	. . . . . . . 8 ⊢ (((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℂ)
72	71	sqcld 14113	. . . . . . 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 25144	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹 supp 0) ⊆ 𝐼)
75	6, 24	rrxsuppss 25144	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐺 supp 0) ⊆ 𝐼)
76	74, 75	unssd 4186	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → ((𝐹 supp 0) ∪ (𝐺 supp 0)) ⊆ 𝐼)
77	76	ssdifssd 4142	. . . . . . . . 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 3982	. . . . . . . 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 4140	. . . . . . . . 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 3982	. . . . . . . 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 4005	. . . . . . . . 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 11211	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → 0 ∈ ℂ)
84	29, 82, 44, 83	suppssr 8183	. . . . . . . 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 2772	. . . . . 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 15675	. . . . 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 481	. . . 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 2776	. . 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 6895	. 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 6906	. 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 7562	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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {crab 3432 Vcvv 3474 ∖ cdif 3945 ∪ cun 3946 ⊆ wss 3948 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5676 Fun wfun 6537 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ∈ cmpo 7413 supp csupp 8148 ↑_m cmap 8822 Fincfn 8941 finSupp cfsupp 9363 ℂcc 11110 ℝcr 11111 0cc0 11112 − cmin 11448 2c2 12271 ↑cexp 14031 √csqrt 15184 Σcsu 15636 Basecbs 17148 distcds 17210 Σ_g cgsu 17390 ℝ_fldcrefld 21376 ℝ^crrx 25124

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-rp 12979 df-fz 13489 df-fzo 13632 df-seq 13971 df-exp 14032 df-hash 14295 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-sum 15637 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-hom 17225 df-cco 17226 df-0g 17391 df-gsum 17392 df-prds 17397 df-pws 17399 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19039 df-ghm 19128 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-cring 20130 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-dvr 20292 df-rhm 20363 df-subrng 20434 df-subrg 20459 df-drng 20502 df-field 20503 df-staf 20596 df-srng 20597 df-lmod 20616 df-lss 20687 df-sra 20930 df-rgmod 20931 df-cnfld 21145 df-refld 21377 df-dsmm 21506 df-frlm 21521 df-nm 24311 df-tng 24313 df-tcph 24910 df-rrx 25126

This theorem is referenced by: rrxmfval 25147 rrxmet 25149 rrxdstprj1 25150

W3C validator