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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 17827 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝐾)) |
8 | 1, 7 | sstrd 3897 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐾)) |
9 | glbsscl.x | . . . . 5 ⊢ (𝜑 → (𝐺‘𝑆) ∈ 𝑇) | |
10 | 8, 9 | sseldd 3888 | . . . 4 ⊢ (𝜑 → (𝐺‘𝑆) ∈ (Base‘𝐾)) |
11 | 5 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝐾 ∈ Poset) |
12 | 6 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑆 ∈ dom 𝐺) |
13 | 1 | sselda 3887 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑆) |
14 | 2, 3, 4, 11, 12, 13 | glble 17832 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → (𝐺‘𝑆)(le‘𝐾)𝑦) |
15 | 14 | ralrimiva 3095 | . . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑇 (𝐺‘𝑆)(le‘𝐾)𝑦) |
16 | breq2 5043 | . . . . . . 7 ⊢ (𝑦 = (𝐺‘𝑆) → (𝑧(le‘𝐾)𝑦 ↔ 𝑧(le‘𝐾)(𝐺‘𝑆))) | |
17 | simp3 1140 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦) → ∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦) | |
18 | 9 | 3ad2ant1 1135 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦) → (𝐺‘𝑆) ∈ 𝑇) |
19 | 16, 17, 18 | rspcdva 3529 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾) ∧ ∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦) → 𝑧(le‘𝐾)(𝐺‘𝑆)) |
20 | 19 | 3expia 1123 | . . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (Base‘𝐾)) → (∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)(𝐺‘𝑆))) |
21 | 20 | ralrimiva 3095 | . . . 4 ⊢ (𝜑 → ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑇 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)(𝐺‘𝑆))) |
22 | breq1 5042 | . . . . . . 7 ⊢ (𝑥 = (𝐺‘𝑆) → (𝑥(le‘𝐾)𝑦 ↔ (𝐺‘𝑆)(le‘𝐾)𝑦)) | |
23 | 22 | ralbidv 3108 | . . . . . 6 ⊢ (𝑥 = (𝐺‘𝑆) → (∀𝑦 ∈ 𝑇 𝑥(le‘𝐾)𝑦 ↔ ∀𝑦 ∈ 𝑇 (𝐺‘𝑆)(le‘𝐾)𝑦)) |
24 | breq2 5043 | . . . . . . . 8 ⊢ (𝑥 = (𝐺‘𝑆) → (𝑧(le‘𝐾)𝑥 ↔ 𝑧(le‘𝐾)(𝐺‘𝑆))) | |
25 | 24 | imbi2d 344 | . . . . . . 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 3108 | . . . . . 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 634 | . . . . 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 3527 | . . . 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 837 | . . 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 264 | . . . 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 45867 | . . 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 17875 | . 2 ⊢ (𝜑 → (𝐺‘𝑇) = (𝐺‘𝑆)) |
36 | 32, 35 | jca 515 | 1 ⊢ (𝜑 → (𝑇 ∈ dom 𝐺 ∧ (𝐺‘𝑇) = (𝐺‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 ⊆ wss 3853 class class class wbr 5039 dom cdm 5536 ‘cfv 6358 Basecbs 16666 lecple 16756 Posetcpo 17768 glbcglb 17771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-dec 12259 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ple 16769 df-odu 17749 df-proset 17756 df-poset 17774 df-lub 17806 df-glb 17807 |
This theorem is referenced by: (None) |
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