Theorem carageneld 41222

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 4306	. . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ dom 𝑂
4	1, 3	syl6eleq 2849	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝒫 ∪ dom 𝑂)
5		simpl 474	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝜑)
6	3	eleq2i 2831	. . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 ↔ 𝑎 ∈ 𝒫 ∪ dom 𝑂)
7	6	bicomi 214	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 ↔ 𝑎 ∈ 𝒫 𝑋)
8	7	biimpi 206	. . . . . 6 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ∈ 𝒫 𝑋)
9	8	adantl 473	. . . . 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 696	. . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
12	11	ralrimiva 3104	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 ∪ dom 𝑂((𝑂‘(𝑎 ∩ 𝐸)) +𝑒 (𝑂‘(𝑎 ∖ 𝐸))) = (𝑂‘𝑎))
13	4, 12	jca 555	. 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 41215	. 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 247	1 ⊢ (𝜑 → 𝐸 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050   ∖ cdif 3712   ∩ cin 3714  𝒫 cpw 4302  ∪ cuni 4588  dom cdm 5266  ‘cfv 6049  (class class class)co 6813   +_𝑒 cxad 12137  OutMeascome 41209  CaraGenccaragen 41211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-caragen 41212

This theorem is referenced by:  caragen0  41226  caragenunidm  41228  caragenuncl  41233  caragendifcl  41234  carageniuncl  41243  caragenel2d  41252