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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 5013 | . . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑘 ∈ 𝐴 𝐵 = ∪ 𝑘 ∈ ∅ 𝐵) | |
2 | 0iun 5068 | . . . . . 6 ⊢ ∪ 𝑘 ∈ ∅ 𝐵 = ∅ | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑘 ∈ ∅ 𝐵 = ∅) |
4 | 1, 3 | eqtrd 2775 | . . . 4 ⊢ (𝐴 = ∅ → ∪ 𝑘 ∈ 𝐴 𝐵 = ∅) |
5 | 4 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → ∪ 𝑘 ∈ 𝐴 𝐵 = ∅) |
6 | caragenfiiuncl.o | . . . . 5 ⊢ (𝜑 → 𝑂 ∈ OutMeas) | |
7 | caragenfiiuncl.s | . . . . 5 ⊢ 𝑆 = (CaraGen‘𝑂) | |
8 | 6, 7 | caragen0 46462 | . . . 4 ⊢ (𝜑 → ∅ ∈ 𝑆) |
9 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → ∅ ∈ 𝑆) |
10 | 5, 9 | eqeltrd 2839 | . 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
11 | simpl 482 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝜑) | |
12 | neqne 2946 | . . . 4 ⊢ (¬ 𝐴 = ∅ → 𝐴 ≠ ∅) | |
13 | 12 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐴 ≠ ∅) |
14 | caragenfiiuncl.kph | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
15 | nfv 1912 | . . . . 5 ⊢ Ⅎ𝑘 𝐴 ≠ ∅ | |
16 | 14, 15 | nfan 1897 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝐴 ≠ ∅) |
17 | caragenfiiuncl.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) | |
18 | 17 | adantlr 715 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ 𝑆) |
19 | 6 | 3ad2ant1 1132 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑂 ∈ OutMeas) |
20 | simp2 1136 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) | |
21 | simp3 1137 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑆) | |
22 | 19, 7, 20, 21 | caragenuncl 46469 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 ∪ 𝑦) ∈ 𝑆) |
23 | 22 | 3adant1r 1176 | . . . 4 ⊢ (((𝜑 ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 ∪ 𝑦) ∈ 𝑆) |
24 | caragenfiiuncl.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
25 | 24 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐴 ∈ Fin) |
26 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → 𝐴 ≠ ∅) | |
27 | 16, 18, 23, 25, 26 | fiiuncl 45005 | . . 3 ⊢ ((𝜑 ∧ 𝐴 ≠ ∅) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
28 | 11, 13, 27 | syl2anc 584 | . 2 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
29 | 10, 28 | pm2.61dan 813 | 1 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 ≠ wne 2938 ∪ cun 3961 ∅c0 4339 ∪ ciun 4996 ‘cfv 6563 Fincfn 8984 OutMeascome 46445 CaraGenccaragen 46447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-xadd 13153 df-icc 13391 df-ome 46446 df-caragen 46448 |
This theorem is referenced by: carageniuncllem1 46477 carageniuncllem2 46478 |
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