Theorem caragensplit 46491

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 45037	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V)
5		ssexg 5273	. . . . . 6 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐴 ∈ V)
6	1, 4, 5	syl2anc 584	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
7		elpwg 4562	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → (𝐴 ∈ 𝒫 𝑋 ↔ 𝐴 ⊆ 𝑋))
9	1, 8	mpbird 257	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝒫 𝑋)
10	3	pweqi 4575	. . 3 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
11	9, 10	eleqtrdi 2838	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 ∪ dom 𝑂)
12		caragensplit.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑆)
13		caragensplit.s	. . . . 5 ⊢ 𝑆 = (CaraGen‘𝑂)
14	2, 13	caragenel 46486	. . . 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 232	. . 3 ⊢ (𝜑 → (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))
16	15	simprd 495	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
17		ineq1 4172	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∩ 𝐸) = (𝐴 ∩ 𝐸))
18	17	fveq2d 6844	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑂‘(𝑎 ∩ 𝐸)) = (𝑂‘(𝐴 ∩ 𝐸)))
19		difeq1 4078	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∖ 𝐸) = (𝐴 ∖ 𝐸))
20	19	fveq2d 6844	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑂‘(𝑎 ∖ 𝐸)) = (𝑂‘(𝐴 ∖ 𝐸)))
21	18, 20	oveq12d 7387	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))))
22		fveq2 6840	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑂‘𝑎) = (𝑂‘𝐴))
23	21, 22	eqeq12d 2745	. . 3 ⊢ (𝑎 = 𝐴 → (((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎) ↔ ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴)))
24	23	rspcva 3583	. 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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3444   ∖ cdif 3908   ∩ cin 3910   ⊆ wss 3911  𝒫 cpw 4559  ∪ cuni 4867  dom cdm 5631  ‘cfv 6499  (class class class)co 7369   +_𝑒 cxad 13046  OutMeascome 46480  CaraGenccaragen 46482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-caragen 46483

This theorem is referenced by:  caragenuncllem  46503  carageniuncllem1  46512  carageniuncllem2  46513  caratheodorylem1  46517