Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 4937 | . . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑘 ∈ 𝐴 𝐵 = ∪ 𝑘 ∈ ∅ 𝐵) | |
2 | 0iun 4988 | . . . . . 6 ⊢ ∪ 𝑘 ∈ ∅ 𝐵 = ∅ | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑘 ∈ ∅ 𝐵 = ∅) |
4 | 1, 3 | eqtrd 2858 | . . . 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 42795 | . . . 4 ⊢ (𝜑 → ∅ ∈ 𝑆) |
9 | 8 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → ∅ ∈ 𝑆) |
10 | 5, 9 | eqeltrd 2915 | . 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
11 | simpl 485 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝜑) | |
12 | neqne 3026 | . . . 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 42802 | . . . . 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 41334 | . . 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 3018 ∪ cun 3936 ∅c0 4293 ∪ ciun 4921 ‘cfv 6357 Fincfn 8511 OutMeascome 42778 CaraGenccaragen 42780 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-1o 8104 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-xadd 12511 df-icc 12748 df-ome 42779 df-caragen 42781 |
This theorem is referenced by: carageniuncllem1 42810 carageniuncllem2 42811 |
Copyright terms: Public domain | W3C validator |