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| Mirrors > Home > MPE Home > Th. List > Mathboxes > glbsscl | Structured version Visualization version GIF version | ||
| Description: If a subset of 𝑆 contains the GLB of 𝑆, then the two sets have the same GLB. (Contributed by Zhi Wang, 26-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| lubsscl.k | ⊢ (𝜑 → 𝐾 ∈ Poset) | 
| lubsscl.t | ⊢ (𝜑 → 𝑇 ⊆ 𝑆) | 
| glbsscl.g | ⊢ 𝐺 = (glb‘𝐾) | 
| glbsscl.s | ⊢ (𝜑 → 𝑆 ∈ dom 𝐺) | 
| glbsscl.x | ⊢ (𝜑 → (𝐺‘𝑆) ∈ 𝑇) | 
| Ref | Expression | 
|---|---|
| glbsscl | ⊢ (𝜑 → (𝑇 ∈ dom 𝐺 ∧ (𝐺‘𝑇) = (𝐺‘𝑆))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | lubsscl.t | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑆) | |
| 2 | eqid 2736 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 3 | eqid 2736 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
| 4 | glbsscl.g | . . . . 5 ⊢ 𝐺 = (glb‘𝐾) | |
| 5 | lubsscl.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Poset) | |
| 6 | glbsscl.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ dom 𝐺) | |
| 7 | 2, 3, 4, 5, 6 | glbelss 18413 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐾)) | 
| 8 | 1, 7 | sstrd 3993 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐾)) | 
| 9 | glbsscl.x | . . . . 5 ⊢ (𝜑 → (𝐺‘𝑆) ∈ 𝑇) | |
| 10 | 8, 9 | sseldd 3983 | . . . 4 ⊢ (𝜑 → (𝐺‘𝑆) ∈ (Base‘𝐾)) | 
| 11 | 5 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝐾 ∈ Poset) | 
| 12 | 6 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑆 ∈ dom 𝐺) | 
| 13 | 1 | sselda 3982 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑆) | 
| 14 | 2, 3, 4, 11, 12, 13 | glble 18418 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → (𝐺‘𝑆)(le‘𝐾)𝑦) | 
| 15 | 14 | ralrimiva 3145 | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑇 (𝐺‘𝑆)(le‘𝐾)𝑦) | 
| 16 | breq2 5146 | . . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑆) → (𝑧(le‘𝐾)𝑦 ↔ 𝑧(le‘𝐾)(𝐺‘𝑆))) | |
| 17 | simp3 1138 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦) → ∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦) | |
| 18 | 9 | 3ad2ant1 1133 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦) → (𝐺‘𝑆) ∈ 𝑇) | 
| 19 | 16, 17, 18 | rspcdva 3622 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦) → 𝑧(le‘𝐾)(𝐺‘𝑆)) | 
| 20 | 19 | 3expia 1121 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾)) → (∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)(𝐺‘𝑆))) | 
| 21 | 20 | ralrimiva 3145 | . . . 4 ⊢ (𝜑 → ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)(𝐺‘𝑆))) | 
| 22 | breq1 5145 | . . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑆) → (𝑥(le‘𝐾)𝑦 ↔ (𝐺‘𝑆)(le‘𝐾)𝑦)) | |
| 23 | 22 | ralbidv 3177 | . . . . . 6 ⊢ (𝑥 = (𝐺‘𝑆) → (∀𝑦 ∈ 𝑇 𝑥(le‘𝐾)𝑦 ↔ ∀𝑦 ∈ 𝑇 (𝐺‘𝑆)(le‘𝐾)𝑦)) | 
| 24 | breq2 5146 | . . . . . . . 8 ⊢ (𝑥 = (𝐺‘𝑆) → (𝑧(le‘𝐾)𝑥 ↔ 𝑧(le‘𝐾)(𝐺‘𝑆))) | |
| 25 | 24 | imbi2d 340 | . . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑆) → ((∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥) ↔ (∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)(𝐺‘𝑆)))) | 
| 26 | 25 | ralbidv 3177 | . . . . . 6 ⊢ (𝑥 = (𝐺‘𝑆) → (∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥) ↔ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)(𝐺‘𝑆)))) | 
| 27 | 23, 26 | anbi12d 632 | . . . . 5 ⊢ (𝑥 = (𝐺‘𝑆) → ((∀𝑦 ∈ 𝑇 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)) ↔ (∀𝑦 ∈ 𝑇 (𝐺‘𝑆)(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)(𝐺‘𝑆))))) | 
| 28 | 27 | rspcev 3621 | . . . 4 ⊢ (((𝐺‘𝑆) ∈ (Base‘𝐾) ∧ (∀𝑦 ∈ 𝑇 (𝐺‘𝑆)(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)(𝐺‘𝑆)))) → ∃𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))) | 
| 29 | 10, 15, 21, 28 | syl12anc 836 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))) | 
| 30 | biid 261 | . . . 4 ⊢ ((∀𝑦 ∈ 𝑇 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)) ↔ (∀𝑦 ∈ 𝑇 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))) | |
| 31 | 2, 3, 4, 30, 5 | glbeldm2 48861 | . . 3 ⊢ (𝜑 → (𝑇 ∈ dom 𝐺 ↔ (𝑇 ⊆ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))))) | 
| 32 | 8, 29, 31 | mpbir2and 713 | . 2 ⊢ (𝜑 → 𝑇 ∈ dom 𝐺) | 
| 33 | eqidd 2737 | . . 3 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐾)) | |
| 34 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐺 = (glb‘𝐾)) | 
| 35 | 3, 33, 34, 5, 8, 10, 14, 19 | posglbdg 18461 | . 2 ⊢ (𝜑 → (𝐺‘𝑇) = (𝐺‘𝑆)) | 
| 36 | 32, 35 | jca 511 | 1 ⊢ (𝜑 → (𝑇 ∈ dom 𝐺 ∧ (𝐺‘𝑇) = (𝐺‘𝑆))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 ⊆ wss 3950 class class class wbr 5142 dom cdm 5684 ‘cfv 6560 Basecbs 17248 lecple 17305 Posetcpo 18354 glbcglb 18357 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-dec 12736 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ple 17318 df-odu 18333 df-proset 18341 df-poset 18360 df-lub 18392 df-glb 18393 | 
| This theorem is referenced by: (None) | 
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