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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 5008 | . . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑘 ∈ 𝐴 𝐵 = ∪ 𝑘 ∈ ∅ 𝐵) | |
| 2 | 0iun 5063 | . . . . . 6 ⊢ ∪ 𝑘 ∈ ∅ 𝐵 = ∅ | |
| 3 | 2 | a1i 11 | . . . . 5 ⊢ (𝐴 = ∅ → ∪ 𝑘 ∈ ∅ 𝐵 = ∅) |
| 4 | 1, 3 | eqtrd 2777 | . . . 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 46521 | . . . 4 ⊢ (𝜑 → ∅ ∈ 𝑆) |
| 9 | 8 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = ∅) → ∅ ∈ 𝑆) |
| 10 | 5, 9 | eqeltrd 2841 | . 2 ⊢ ((𝜑 ∧ 𝐴 = ∅) → ∪ 𝑘 ∈ 𝐴 𝐵 ∈ 𝑆) |
| 11 | simpl 482 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝜑) | |
| 12 | neqne 2948 | . . . 4 ⊢ (¬ 𝐴 = ∅ → 𝐴 ≠ ∅) | |
| 13 | 12 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = ∅) → 𝐴 ≠ ∅) |
| 14 | caragenfiiuncl.kph | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
| 15 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑘 𝐴 ≠ ∅ | |
| 16 | 14, 15 | nfan 1899 | . . . 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 1134 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑂 ∈ OutMeas) |
| 20 | simp2 1138 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) | |
| 21 | simp3 1139 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑆) | |
| 22 | 19, 7, 20, 21 | caragenuncl 46528 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 ∪ 𝑦) ∈ 𝑆) |
| 23 | 22 | 3adant1r 1178 | . . . 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 45070 | . . 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 1087 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ≠ wne 2940 ∪ cun 3949 ∅c0 4333 ∪ ciun 4991 ‘cfv 6561 Fincfn 8985 OutMeascome 46504 CaraGenccaragen 46506 |
| 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-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-pss 3971 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-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 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-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 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-om 7888 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-xadd 13155 df-icc 13394 df-ome 46505 df-caragen 46507 |
| This theorem is referenced by: carageniuncllem1 46536 carageniuncllem2 46537 |
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