Theorem carageneld 46493

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 4581	. . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
4	1, 3	eleqtrdi 2839	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝒫 ∪ dom 𝑂)
5		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝜑)
6	3	eleq2i 2821	. . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 ↔ 𝑎 ∈ 𝒫 ∪ dom 𝑂)
7	6	bicomi 224	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 ↔ 𝑎 ∈ 𝒫 𝑋)
8	7	biimpi 216	. . . . . 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 584	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
12	11	ralrimiva 3126	. . 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 46486	. 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 257	1 ⊢ (𝜑 → 𝐸 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045   ∖ cdif 3913   ∩ cin 3915  𝒫 cpw 4565  ∪ cuni 4873  dom cdm 5640  ‘cfv 6513  (class class class)co 7389   +_𝑒 cxad 13076  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 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-iota 6466  df-fun 6515  df-fv 6521  df-ov 7392  df-caragen 46483

This theorem is referenced by:  caragen0  46497  caragenunidm  46499  caragenuncl  46504  caragendifcl  46505  carageniuncl  46514  caragenel2d  46523