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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > caragenfiiuncl | Structured version Visualization version GIF version |
Description: The Caratheodory's construction is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
caragenfiiuncl.kph | ⊢ Ⅎ𝑘𝜑 |
caragenfiiuncl.o | ⊢ (𝜑 → 𝑂 ∈ OutMeas) |
caragenfiiuncl.s | ⊢ 𝑆 = (CaraGen‘𝑂) |
caragenfiiuncl.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
caragenfiiuncl.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) |
Ref | Expression |
---|---|
caragenfiiuncl | ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iuneq1 4801 | . . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑘 ∈ 𝐴 𝐵 = ∪ 𝑘 ∈ ∅ 𝐵) | |
2 | 0iun 4846 | . . . . . 6 ⊢ ∪ 𝑘 ∈ ∅ 𝐵 = ∅ | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑘 ∈ ∅ 𝐵 = ∅) |
4 | 1, 3 | eqtrd 2808 | . . . 4 ⊢ (𝐴 = ∅ → ∪ 𝑘 ∈ 𝐴 𝐵 = ∅) |
5 | 4 | adantl 474 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → ∪ 𝑘 ∈ 𝐴 𝐵 = ∅) |
6 | caragenfiiuncl.o | . . . . 5 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
7 | caragenfiiuncl.s | . . . . 5 ⊢ 𝑆 = (CaraGen‘𝑂) | |
8 | 6, 7 | caragen0 42219 | . . . 4 ⊢ (𝜑 → ∅ ∈ 𝑆) |
9 | 8 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → ∅ ∈ 𝑆) |
10 | 5, 9 | eqeltrd 2860 | . 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
11 | simpl 475 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝜑) | |
12 | neqne 2969 | . . . 4 ⊢ (¬ 𝐴 = ∅ → 𝐴 ≠ ∅) | |
13 | 12 | adantl 474 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐴 ≠ ∅) |
14 | caragenfiiuncl.kph | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
15 | nfv 1873 | . . . . 5 ⊢ Ⅎ𝑘 𝐴 ≠ ∅ | |
16 | 14, 15 | nfan 1862 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐴 ≠ ∅) |
17 | caragenfiiuncl.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) | |
18 | 17 | adantlr 702 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) |
19 | 6 | 3ad2ant1 1113 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑂 ∈ OutMeas) |
20 | simp2 1117 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) | |
21 | simp3 1118 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑆) | |
22 | 19, 7, 20, 21 | caragenuncl 42226 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 ∪ 𝑦) ∈ 𝑆) |
23 | 22 | 3adant1r 1157 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 ∪ 𝑦) ∈ 𝑆) |
24 | caragenfiiuncl.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
25 | 24 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Fin) |
26 | simpr 477 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅) | |
27 | 16, 18, 23, 25, 26 | fiiuncl 40746 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
28 | 11, 13, 27 | syl2anc 576 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
29 | 10, 28 | pm2.61dan 800 | 1 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 Ⅎwnf 1746 ∈ wcel 2050 ≠ wne 2961 ∪ cun 3821 ∅c0 4172 ∪ ciun 4786 ‘cfv 6182 Fincfn 8300 OutMeascome 42202 CaraGenccaragen 42204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-1o 7899 df-er 8083 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-xadd 12319 df-icc 12555 df-ome 42203 df-caragen 42205 |
This theorem is referenced by: carageniuncllem1 42234 carageniuncllem2 42235 |
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