Theorem carageneld 43994

Description: Membership in the Caratheodory's construction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
carageneld.o	⊢ (𝜑 → 𝑂 ∈ OutMeas)
carageneld.x	⊢ 𝑋 = ∪ dom 𝑂
carageneld.s	⊢ 𝑆 = (CaraGen‘𝑂)
carageneld.e	⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑋)
carageneld.a	⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))

Assertion

Ref	Expression
carageneld	⊢ (𝜑 → 𝐸 ∈ 𝑆)

Distinct variable groups:   𝐸,𝑎   𝑂,𝑎   𝜑,𝑎

Allowed substitution hints:   𝑆(𝑎)   𝑋(𝑎)

Proof of Theorem carageneld

Step	Hyp	Ref	Expression
1		carageneld.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝒫 𝑋)
2		carageneld.x	. . . . 5 ⊢ 𝑋 = ∪ dom 𝑂
3	2	pweqi 4556	. . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
4	1, 3	eleqtrdi 2850	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝒫 ∪ dom 𝑂)
5		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝜑)
6	3	eleq2i 2831	. . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 ↔ 𝑎 ∈ 𝒫 ∪ dom 𝑂)
7	6	bicomi 223	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 ↔ 𝑎 ∈ 𝒫 𝑋)
8	7	biimpi 215	. . . . . 6 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ∈ 𝒫 𝑋)
9	8	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ∈ 𝒫 𝑋)
10		carageneld.a	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
11	5, 9, 10	syl2anc 583	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
12	11	ralrimiva 3109	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
13	4, 12	jca 511	. 2 ⊢ (𝜑 → (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎)))
14		carageneld.o	. . 3 ⊢ (𝜑 → 𝑂 ∈ OutMeas)
15		carageneld.s	. . 3 ⊢ 𝑆 = (CaraGen‘𝑂)
16	14, 15	caragenel 43987	. 2 ⊢ (𝜑 → (𝐸 ∈ 𝑆 ↔ (𝐸 ∈ 𝒫 ∪ dom 𝑂 ∧ ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))))
17	13, 16	mpbird 256	1 ⊢ (𝜑 → 𝐸 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109  ∀wral 3065   ∖ cdif 3888   ∩ cin 3890  𝒫 cpw 4538  ∪ cuni 4844  dom cdm 5588  ‘cfv 6430  (class class class)co 7268   +_𝑒 cxad 12828  OutMeascome 43981  CaraGenccaragen 43983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-iota 6388  df-fun 6432  df-fv 6438  df-ov 7271  df-caragen 43984

This theorem is referenced by:  caragen0  43998  caragenunidm  44000  caragenuncl  44005  caragendifcl  44006  carageniuncl  44015  caragenel2d  44024