Theorem caragensplit 45801

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 44327	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V)
5		ssexg 5317	. . . . . 6 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑋 ∈ V) → 𝐴 ∈ V)
6	1, 4, 5	syl2anc 583	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
7		elpwg 4601	. . . . 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 4614	. . 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 45796	. . . 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 495	. 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
17		ineq1 4201	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∩ 𝐸) = (𝐴 ∩ 𝐸))
18	17	fveq2d 6895	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑂‘(𝑎 ∩ 𝐸)) = (𝑂‘(𝐴 ∩ 𝐸)))
19		difeq1 4111	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 ∖ 𝐸) = (𝐴 ∖ 𝐸))
20	19	fveq2d 6895	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑂‘(𝑎 ∖ 𝐸)) = (𝑂‘(𝐴 ∖ 𝐸)))
21	18, 20	oveq12d 7432	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))))
22		fveq2 6891	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑂‘𝑎) = (𝑂‘𝐴))
23	21, 22	eqeq12d 2743	. . 3 ⊢ (𝑎 = 𝐴 → (((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎) ↔ ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴)))
24	23	rspcva 3605	. 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 583	1 ⊢ (𝜑 → ((𝑂‘(𝐴 ∩ 𝐸)) +𝑒 (𝑂‘(𝐴 ∖ 𝐸))) = (𝑂‘𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3056  Vcvv 3469   ∖ cdif 3941   ∩ cin 3943   ⊆ wss 3944  𝒫 cpw 4598  ∪ cuni 4903  dom cdm 5672  ‘cfv 6542  (class class class)co 7414   +_𝑒 cxad 13108  OutMeascome 45790  CaraGenccaragen 45792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-caragen 45793

This theorem is referenced by:  caragenuncllem  45813  carageniuncllem1  45822  carageniuncllem2  45823  caratheodorylem1  45827