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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 4935 | . . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑘 ∈ 𝐴 𝐵 = ∪ 𝑘 ∈ ∅ 𝐵) | |
2 | 0iun 4986 | . . . . . 6 ⊢ ∪ 𝑘 ∈ ∅ 𝐵 = ∅ | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑘 ∈ ∅ 𝐵 = ∅) |
4 | 1, 3 | eqtrd 2856 | . . . 4 ⊢ (𝐴 = ∅ → ∪ 𝑘 ∈ 𝐴 𝐵 = ∅) |
5 | 4 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → ∪ 𝑘 ∈ 𝐴 𝐵 = ∅) |
6 | caragenfiiuncl.o | . . . . 5 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
7 | caragenfiiuncl.s | . . . . 5 ⊢ 𝑆 = (CaraGen‘𝑂) | |
8 | 6, 7 | caragen0 42808 | . . . 4 ⊢ (𝜑 → ∅ ∈ 𝑆) |
9 | 8 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → ∅ ∈ 𝑆) |
10 | 5, 9 | eqeltrd 2913 | . 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
11 | simpl 485 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝜑) | |
12 | neqne 3024 | . . . 4 ⊢ (¬ 𝐴 = ∅ → 𝐴 ≠ ∅) | |
13 | 12 | adantl 484 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐴 ≠ ∅) |
14 | caragenfiiuncl.kph | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
15 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑘 𝐴 ≠ ∅ | |
16 | 14, 15 | nfan 1900 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐴 ≠ ∅) |
17 | caragenfiiuncl.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) | |
18 | 17 | adantlr 713 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) |
19 | 6 | 3ad2ant1 1129 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑂 ∈ OutMeas) |
20 | simp2 1133 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) | |
21 | simp3 1134 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑆) | |
22 | 19, 7, 20, 21 | caragenuncl 42815 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 ∪ 𝑦) ∈ 𝑆) |
23 | 22 | 3adant1r 1173 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 ∪ 𝑦) ∈ 𝑆) |
24 | caragenfiiuncl.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
25 | 24 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Fin) |
26 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅) | |
27 | 16, 18, 23, 25, 26 | fiiuncl 41347 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
28 | 11, 13, 27 | syl2anc 586 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
29 | 10, 28 | pm2.61dan 811 | 1 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 ≠ wne 3016 ∪ cun 3934 ∅c0 4291 ∪ ciun 4919 ‘cfv 6355 Fincfn 8509 OutMeascome 42791 CaraGenccaragen 42793 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-1o 8102 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-xadd 12509 df-icc 12746 df-ome 42792 df-caragen 42794 |
This theorem is referenced by: carageniuncllem1 42823 carageniuncllem2 42824 |
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