Theorem caragensplit 45202

Description: If 𝐸 is in the set generated by the Caratheodory's method, then it splits any set 𝐴 in two parts such that the sum of the outer measures of the two parts is equal to the outer measure of the whole set 𝐴. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
caragensplit.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
caragensplit.s	⊢ 𝑆 = (CaraGen‘𝑂)
caragensplit.x	⊢ 𝑋 = ∪ dom 𝑂
caragensplit.e	⊢ (𝜑 → 𝐸 ∈ 𝑆)
caragensplit.a	⊢ (𝜑 → 𝐴 ⊆ 𝑋)

Assertion

Ref	Expression
caragensplit	⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴))

Proof of Theorem caragensplit

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caragensplit.a	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑋)
2		caragensplit.o	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
3		caragensplit.x	. . . . . . 7 ⊢ 𝑋 = ∪ dom 𝑂
4	2, 3	unidmex 43722	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V)
5		ssexg 5322	. . . . . 6 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐴 ∈ V)
6	1, 4, 5	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
7		elpwg 4604	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
9	1, 8	mpbird 256	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋)
10	3	pweqi 4617	. . 3 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
11	9, 10	eleqtrdi 2843	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 ∪ dom 𝑂)
12		caragensplit.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑆)
13		caragensplit.s	. . . . 5 ⊢ 𝑆 = (CaraGen‘𝑂)
14	2, 13	caragenel 45197	. . . 4 ⊢ (𝜑 → (𝐸 ∈ 𝑆 ↔ (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))))
15	12, 14	mpbid 231	. . 3 ⊢ (𝜑 → (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))
16	15	simprd 496	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
17		ineq1 4204	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∩ 𝐸) = (𝐴 ∩ 𝐸))
18	17	fveq2d 6892	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑂‘(𝑎 ∩ 𝐸)) = (𝑂‘(𝐴 ∩ 𝐸)))
19		difeq1 4114	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∖ 𝐸) = (𝐴 ∖ 𝐸))
20	19	fveq2d 6892	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑂‘(𝑎 ∖ 𝐸)) = (𝑂‘(𝐴 ∖ 𝐸)))
21	18, 20	oveq12d 7423	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))))
22		fveq2 6888	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑂‘𝑎) = (𝑂‘𝐴))
23	21, 22	eqeq12d 2748	. . 3 ⊢ (𝑎 = 𝐴 → (((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎) ↔ ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴)))
24	23	rspcva 3610	. 2 ⊢ ((𝐴 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)) → ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴))
25	11, 16, 24	syl2anc 584	1 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  Vcvv 3474   ∖ cdif 3944   ∩ cin 3946   ⊆ wss 3947  𝒫 cpw 4601  ∪ cuni 4907  dom cdm 5675  ‘cfv 6540  (class class class)co 7405   +_𝑒 cxad 13086  OutMeascome 45191  CaraGenccaragen 45193

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-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-caragen 45194

This theorem is referenced by:  caragenuncllem  45214  carageniuncllem1  45223  carageniuncllem2  45224  caratheodorylem1  45228