Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > caragen0 | Structured version Visualization version GIF version |
Description: The empty set belongs to any Caratheodory's construction. First part of Step (b) in the proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
caragen0.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
caragen0.s | ⊢ 𝑆 = (CaraGen‘𝑂) |
Ref | Expression |
---|---|
caragen0 | ⊢ (𝜑 → ∅ ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caragen0.o | . 2 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
2 | eqid 2622 | . 2 ⊢ ∪ dom 𝑂 = ∪ dom 𝑂 | |
3 | caragen0.s | . 2 ⊢ 𝑆 = (CaraGen‘𝑂) | |
4 | 0elpw 4834 | . . 3 ⊢ ∅ ∈ 𝒫 ∪ dom 𝑂 | |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → ∅ ∈ 𝒫 ∪ dom 𝑂) |
6 | in0 3968 | . . . . . 6 ⊢ (𝑎 ∩ ∅) = ∅ | |
7 | 6 | fveq2i 6194 | . . . . 5 ⊢ (𝑂‘(𝑎 ∩ ∅)) = (𝑂‘∅) |
8 | dif0 3950 | . . . . . 6 ⊢ (𝑎 ∖ ∅) = 𝑎 | |
9 | 8 | fveq2i 6194 | . . . . 5 ⊢ (𝑂‘(𝑎 ∖ ∅)) = (𝑂‘𝑎) |
10 | 7, 9 | oveq12i 6662 | . . . 4 ⊢ ((𝑂‘(𝑎 ∩ ∅)) +𝑒 (𝑂‘(𝑎 ∖ ∅))) = ((𝑂‘∅) +𝑒 (𝑂‘𝑎)) |
11 | 10 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ ∅)) +𝑒 (𝑂‘(𝑎 ∖ ∅))) = ((𝑂‘∅) +𝑒 (𝑂‘𝑎))) |
12 | 1 | ome0 40711 | . . . . 5 ⊢ (𝜑 → (𝑂‘∅) = 0) |
13 | 12 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘∅) = 0) |
14 | 13 | oveq1d 6665 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘∅) +𝑒 (𝑂‘𝑎)) = (0 +𝑒 (𝑂‘𝑎))) |
15 | iccssxr 12256 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
16 | 1 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑂 ∈ OutMeas) |
17 | elpwi 4168 | . . . . . . 7 ⊢ (𝑎 ∈ 𝒫 ∪ dom 𝑂 → 𝑎 ⊆ ∪ dom 𝑂) | |
18 | 17 | adantl 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → 𝑎 ⊆ ∪ dom 𝑂) |
19 | 16, 2, 18 | omecl 40717 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘𝑎) ∈ (0[,]+∞)) |
20 | 15, 19 | sseldi 3601 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (𝑂‘𝑎) ∈ ℝ*) |
21 | 20 | xaddid2d 39535 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → (0 +𝑒 (𝑂‘𝑎)) = (𝑂‘𝑎)) |
22 | 11, 14, 21 | 3eqtrd 2660 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝒫 ∪ dom 𝑂) → ((𝑂‘(𝑎 ∩ ∅)) +𝑒 (𝑂‘(𝑎 ∖ ∅))) = (𝑂‘𝑎)) |
23 | 1, 2, 3, 5, 22 | carageneld 40716 | 1 ⊢ (𝜑 → ∅ ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 ∪ cuni 4436 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 0cc0 9936 +∞cpnf 10071 ℝ*cxr 10073 +𝑒 cxad 11944 [,]cicc 12178 OutMeascome 40703 CaraGenccaragen 40705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-xadd 11947 df-icc 12182 df-ome 40704 df-caragen 40706 |
This theorem is referenced by: caragenfiiuncl 40729 caragenunicl 40738 caragensal 40739 caratheodory 40742 |
Copyright terms: Public domain | W3C validator |