Theorem caragensplit 46811

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 45362	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V)
5		ssexg 5269	. . . . . 6 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐴 ∈ V)
6	1, 4, 5	syl2anc 585	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
7		elpwg 4558	. . . . 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 4571	. . 3 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
11	9, 10	eleqtrdi 2847	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 ∪ dom 𝑂)
12		caragensplit.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑆)
13		caragensplit.s	. . . . 5 ⊢ 𝑆 = (CaraGen‘𝑂)
14	2, 13	caragenel 46806	. . . 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 4166	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∩ 𝐸) = (𝐴 ∩ 𝐸))
18	17	fveq2d 6839	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑂‘(𝑎 ∩ 𝐸)) = (𝑂‘(𝐴 ∩ 𝐸)))
19		difeq1 4072	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∖ 𝐸) = (𝐴 ∖ 𝐸))
20	19	fveq2d 6839	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑂‘(𝑎 ∖ 𝐸)) = (𝑂‘(𝐴 ∖ 𝐸)))
21	18, 20	oveq12d 7378	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))))
22		fveq2 6835	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑂‘𝑎) = (𝑂‘𝐴))
23	21, 22	eqeq12d 2753	. . 3 ⊢ (𝑎 = 𝐴 → (((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎) ↔ ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴)))
24	23	rspcva 3575	. 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 585	1 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3441   ∖ cdif 3899   ∩ cin 3901   ⊆ wss 3902  𝒫 cpw 4555  ∪ cuni 4864  dom cdm 5625  ‘cfv 6493  (class class class)co 7360   +_𝑒 cxad 13028  OutMeascome 46800  CaraGenccaragen 46802

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-iota 6449  df-fun 6495  df-fv 6501  df-ov 7363  df-caragen 46803

This theorem is referenced by:  caragenuncllem  46823  carageniuncllem1  46832  carageniuncllem2  46833  caratheodorylem1  46837