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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ipolub | Structured version Visualization version GIF version | ||
| Description: The LUB of the inclusion poset. (hypotheses "ipolub.s" and "ipolub.t" could be eliminated with 𝑆 ∈ dom 𝑈.) Could be significantly shortened if poslubdg 18459 is in quantified form. mrelatlub 18607 could potentially be shortened using this. See mrelatlubALT 48884. (Contributed by Zhi Wang, 28-Sep-2024.) |
| Ref | Expression |
|---|---|
| ipolub.i | ⊢ 𝐼 = (toInc‘𝐹) |
| ipolub.f | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
| ipolub.s | ⊢ (𝜑 → 𝑆 ⊆ 𝐹) |
| ipolub.u | ⊢ (𝜑 → 𝑈 = (lub‘𝐼)) |
| ipolubdm.t | ⊢ (𝜑 → 𝑇 = ∩ {𝑥 ∈ 𝐹 ∣ ∪ 𝑆 ⊆ 𝑥}) |
| ipolub.t | ⊢ (𝜑 → 𝑇 ∈ 𝐹) |
| Ref | Expression |
|---|---|
| ipolub | ⊢ (𝜑 → (𝑈‘𝑆) = 𝑇) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . 2 ⊢ (le‘𝐼) = (le‘𝐼) | |
| 2 | ipolub.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
| 3 | ipolub.i | . . . 4 ⊢ 𝐼 = (toInc‘𝐹) | |
| 4 | 3 | ipobas 18576 | . . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐹 = (Base‘𝐼)) |
| 5 | 2, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝐹 = (Base‘𝐼)) |
| 6 | ipolub.u | . 2 ⊢ (𝜑 → 𝑈 = (lub‘𝐼)) | |
| 7 | 3 | ipopos 18581 | . . 3 ⊢ 𝐼 ∈ Poset |
| 8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → 𝐼 ∈ Poset) |
| 9 | ipolub.s | . 2 ⊢ (𝜑 → 𝑆 ⊆ 𝐹) | |
| 10 | ipolub.t | . 2 ⊢ (𝜑 → 𝑇 ∈ 𝐹) | |
| 11 | breq1 5146 | . . 3 ⊢ (𝑤 = 𝑦 → (𝑤(le‘𝐼)𝑇 ↔ 𝑦(le‘𝐼)𝑇)) | |
| 12 | ipolubdm.t | . . . . . . 7 ⊢ (𝜑 → 𝑇 = ∩ {𝑥 ∈ 𝐹 ∣ ∪ 𝑆 ⊆ 𝑥}) | |
| 13 | intubeu 48873 | . . . . . . . 8 ⊢ (𝑇 ∈ 𝐹 → ((∪ 𝑆 ⊆ 𝑇 ∧ ∀𝑣 ∈ 𝐹 (∪ 𝑆 ⊆ 𝑣 → 𝑇 ⊆ 𝑣)) ↔ 𝑇 = ∩ {𝑥 ∈ 𝐹 ∣ ∪ 𝑆 ⊆ 𝑥})) | |
| 14 | 13 | biimpar 477 | . . . . . . 7 ⊢ ((𝑇 ∈ 𝐹 ∧ 𝑇 = ∩ {𝑥 ∈ 𝐹 ∣ ∪ 𝑆 ⊆ 𝑥}) → (∪ 𝑆 ⊆ 𝑇 ∧ ∀𝑣 ∈ 𝐹 (∪ 𝑆 ⊆ 𝑣 → 𝑇 ⊆ 𝑣))) |
| 15 | 10, 12, 14 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (∪ 𝑆 ⊆ 𝑇 ∧ ∀𝑣 ∈ 𝐹 (∪ 𝑆 ⊆ 𝑣 → 𝑇 ⊆ 𝑣))) |
| 16 | 3, 2, 9, 1 | ipolublem 48875 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 ∈ 𝐹) → ((∪ 𝑆 ⊆ 𝑇 ∧ ∀𝑣 ∈ 𝐹 (∪ 𝑆 ⊆ 𝑣 → 𝑇 ⊆ 𝑣)) ↔ (∀𝑤 ∈ 𝑆 𝑤(le‘𝐼)𝑇 ∧ ∀𝑣 ∈ 𝐹 (∀𝑤 ∈ 𝑆 𝑤(le‘𝐼)𝑣 → 𝑇(le‘𝐼)𝑣)))) |
| 17 | 10, 16 | mpdan 687 | . . . . . 6 ⊢ (𝜑 → ((∪ 𝑆 ⊆ 𝑇 ∧ ∀𝑣 ∈ 𝐹 (∪ 𝑆 ⊆ 𝑣 → 𝑇 ⊆ 𝑣)) ↔ (∀𝑤 ∈ 𝑆 𝑤(le‘𝐼)𝑇 ∧ ∀𝑣 ∈ 𝐹 (∀𝑤 ∈ 𝑆 𝑤(le‘𝐼)𝑣 → 𝑇(le‘𝐼)𝑣)))) |
| 18 | 15, 17 | mpbid 232 | . . . . 5 ⊢ (𝜑 → (∀𝑤 ∈ 𝑆 𝑤(le‘𝐼)𝑇 ∧ ∀𝑣 ∈ 𝐹 (∀𝑤 ∈ 𝑆 𝑤(le‘𝐼)𝑣 → 𝑇(le‘𝐼)𝑣))) |
| 19 | 18 | simpld 494 | . . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝑆 𝑤(le‘𝐼)𝑇) |
| 20 | 19 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → ∀𝑤 ∈ 𝑆 𝑤(le‘𝐼)𝑇) |
| 21 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ 𝑆) | |
| 22 | 11, 20, 21 | rspcdva 3623 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦(le‘𝐼)𝑇) |
| 23 | breq2 5147 | . . . . . . 7 ⊢ (𝑣 = 𝑧 → (𝑤(le‘𝐼)𝑣 ↔ 𝑤(le‘𝐼)𝑧)) | |
| 24 | 23 | ralbidv 3178 | . . . . . 6 ⊢ (𝑣 = 𝑧 → (∀𝑤 ∈ 𝑆 𝑤(le‘𝐼)𝑣 ↔ ∀𝑤 ∈ 𝑆 𝑤(le‘𝐼)𝑧)) |
| 25 | breq1 5146 | . . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤(le‘𝐼)𝑧 ↔ 𝑦(le‘𝐼)𝑧)) | |
| 26 | 25 | cbvralvw 3237 | . . . . . 6 ⊢ (∀𝑤 ∈ 𝑆 𝑤(le‘𝐼)𝑧 ↔ ∀𝑦 ∈ 𝑆 𝑦(le‘𝐼)𝑧) |
| 27 | 24, 26 | bitrdi 287 | . . . . 5 ⊢ (𝑣 = 𝑧 → (∀𝑤 ∈ 𝑆 𝑤(le‘𝐼)𝑣 ↔ ∀𝑦 ∈ 𝑆 𝑦(le‘𝐼)𝑧)) |
| 28 | breq2 5147 | . . . . 5 ⊢ (𝑣 = 𝑧 → (𝑇(le‘𝐼)𝑣 ↔ 𝑇(le‘𝐼)𝑧)) | |
| 29 | 27, 28 | imbi12d 344 | . . . 4 ⊢ (𝑣 = 𝑧 → ((∀𝑤 ∈ 𝑆 𝑤(le‘𝐼)𝑣 → 𝑇(le‘𝐼)𝑣) ↔ (∀𝑦 ∈ 𝑆 𝑦(le‘𝐼)𝑧 → 𝑇(le‘𝐼)𝑧))) |
| 30 | 18 | simprd 495 | . . . . 5 ⊢ (𝜑 → ∀𝑣 ∈ 𝐹 (∀𝑤 ∈ 𝑆 𝑤(le‘𝐼)𝑣 → 𝑇(le‘𝐼)𝑣)) |
| 31 | 30 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐹) → ∀𝑣 ∈ 𝐹 (∀𝑤 ∈ 𝑆 𝑤(le‘𝐼)𝑣 → 𝑇(le‘𝐼)𝑣)) |
| 32 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐹) → 𝑧 ∈ 𝐹) | |
| 33 | 29, 31, 32 | rspcdva 3623 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐹) → (∀𝑦 ∈ 𝑆 𝑦(le‘𝐼)𝑧 → 𝑇(le‘𝐼)𝑧)) |
| 34 | 33 | 3impia 1118 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐹 ∧ ∀𝑦 ∈ 𝑆 𝑦(le‘𝐼)𝑧) → 𝑇(le‘𝐼)𝑧) |
| 35 | 1, 5, 6, 8, 9, 10, 22, 34 | poslubdg 18459 | 1 ⊢ (𝜑 → (𝑈‘𝑆) = 𝑇) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 ⊆ wss 3951 ∪ cuni 4907 ∩ cint 4946 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 lecple 17304 Posetcpo 18353 lubclub 18355 toInccipo 18572 |
| 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-rep 5279 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 ax-pre-mulgt0 11232 |
| 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-rmo 3380 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-int 4947 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-pred 6321 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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 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-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-tset 17316 df-ple 17317 df-ocomp 17318 df-proset 18340 df-poset 18359 df-lub 18391 df-ipo 18573 |
| This theorem is referenced by: ipolub0 48881 mrelatlubALT 48884 toplatlub 48889 toplatjoin 48891 |
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