Theorem carageneld 46688

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 4568	. . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
4	1, 3	eleqtrdi 2844	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝒫 ∪ dom 𝑂)
5		simpl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝜑)
6	3	eleq2i 2826	. . . . . . . 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 46681	. 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 1541   ∈ wcel 2113  ∀wral 3049   ∖ cdif 3896   ∩ cin 3898  𝒫 cpw 4552  ∪ cuni 4861  dom cdm 5622  ‘cfv 6490  (class class class)co 7356   +_𝑒 cxad 13022  OutMeascome 46675  CaraGenccaragen 46677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7359  df-caragen 46678

This theorem is referenced by:  caragen0  46692  caragenunidm  46694  caragenuncl  46699  caragendifcl  46700  carageniuncl  46709  caragenel2d  46718