| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > 0elcarsg | Structured version Visualization version GIF version | ||
| Description: The empty set is Caratheodory measurable. (Contributed by Thierry Arnoux, 30-May-2020.) |
| Ref | Expression |
|---|---|
| carsgval.1 | ⊢ (𝜑 → 𝑂 ∈ 𝑉) |
| carsgval.2 | ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) |
| baselcarsg.1 | ⊢ (𝜑 → (𝑀‘∅) = 0) |
| Ref | Expression |
|---|---|
| 0elcarsg | ⊢ (𝜑 → ∅ ∈ (toCaraSiga‘𝑀)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0ss 4400 | . . 3 ⊢ ∅ ⊆ 𝑂 | |
| 2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ∅ ⊆ 𝑂) |
| 3 | in0 4395 | . . . . . . . 8 ⊢ (𝑒 ∩ ∅) = ∅ | |
| 4 | 3 | fveq2i 6909 | . . . . . . 7 ⊢ (𝑀‘(𝑒 ∩ ∅)) = (𝑀‘∅) |
| 5 | baselcarsg.1 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘∅) = 0) | |
| 6 | 4, 5 | eqtrid 2789 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑒 ∩ ∅)) = 0) |
| 7 | dif0 4378 | . . . . . . . 8 ⊢ (𝑒 ∖ ∅) = 𝑒 | |
| 8 | 7 | fveq2i 6909 | . . . . . . 7 ⊢ (𝑀‘(𝑒 ∖ ∅)) = (𝑀‘𝑒) |
| 9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑒 ∖ ∅)) = (𝑀‘𝑒)) |
| 10 | 6, 9 | oveq12d 7449 | . . . . 5 ⊢ (𝜑 → ((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (0 +𝑒 (𝑀‘𝑒))) |
| 11 | 10 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (0 +𝑒 (𝑀‘𝑒))) |
| 12 | iccssxr 13470 | . . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ* | |
| 13 | carsgval.2 | . . . . . . 7 ⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞)) | |
| 14 | 13 | ffvelcdmda 7104 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘𝑒) ∈ (0[,]+∞)) |
| 15 | 12, 14 | sselid 3981 | . . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (𝑀‘𝑒) ∈ ℝ*) |
| 16 | xaddlid 13284 | . . . . 5 ⊢ ((𝑀‘𝑒) ∈ ℝ* → (0 +𝑒 (𝑀‘𝑒)) = (𝑀‘𝑒)) | |
| 17 | 15, 16 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → (0 +𝑒 (𝑀‘𝑒)) = (𝑀‘𝑒)) |
| 18 | 11, 17 | eqtrd 2777 | . . 3 ⊢ ((𝜑 ∧ 𝑒 ∈ 𝒫 𝑂) → ((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (𝑀‘𝑒)) |
| 19 | 18 | ralrimiva 3146 | . 2 ⊢ (𝜑 → ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (𝑀‘𝑒)) |
| 20 | carsgval.1 | . . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑉) | |
| 21 | 20, 13 | elcarsg 34307 | . 2 ⊢ (𝜑 → (∅ ∈ (toCaraSiga‘𝑀) ↔ (∅ ⊆ 𝑂 ∧ ∀𝑒 ∈ 𝒫 𝑂((𝑀‘(𝑒 ∩ ∅)) +𝑒 (𝑀‘(𝑒 ∖ ∅))) = (𝑀‘𝑒)))) |
| 22 | 2, 19, 21 | mpbir2and 713 | 1 ⊢ (𝜑 → ∅ ∈ (toCaraSiga‘𝑀)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 0cc0 11155 +∞cpnf 11292 ℝ*cxr 11294 +𝑒 cxad 13152 [,]cicc 13390 toCaraSigaccarsg 34303 |
| 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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-xadd 13155 df-icc 13394 df-carsg 34304 |
| This theorem is referenced by: carsggect 34320 omsmeas 34325 |
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