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Mirrors > Home > MPE Home > Th. List > glbcl | Structured version Visualization version GIF version |
Description: The least upper bound function value belongs to the base set. (Contributed by NM, 7-Sep-2018.) |
Ref | Expression |
---|---|
glbc.b | ⊢ 𝐵 = (Base‘𝐾) |
glbc.g | ⊢ 𝐺 = (glb‘𝐾) |
glbc.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
glbc.s | ⊢ (𝜑 → 𝑆 ∈ dom 𝐺) |
Ref | Expression |
---|---|
glbcl | ⊢ (𝜑 → (𝐺‘𝑆) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | glbc.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2823 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | glbc.g | . . 3 ⊢ 𝐺 = (glb‘𝐾) | |
4 | biid 263 | . . 3 ⊢ ((∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)) ↔ (∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))) | |
5 | glbc.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
6 | glbc.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ dom 𝐺) | |
7 | 1, 2, 3, 5, 6 | glbelss 17607 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
8 | 1, 2, 3, 4, 5, 7 | glbval 17609 | . 2 ⊢ (𝜑 → (𝐺‘𝑆) = (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))) |
9 | 1, 2, 3, 4, 5, 6 | glbeu 17608 | . . 3 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))) |
10 | riotacl 7133 | . . 3 ⊢ (∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)) → (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))) ∈ 𝐵) | |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))) ∈ 𝐵) |
12 | 8, 11 | eqeltrd 2915 | 1 ⊢ (𝜑 → (𝐺‘𝑆) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃!wreu 3142 class class class wbr 5068 dom cdm 5557 ‘cfv 6357 ℩crio 7115 Basecbs 16485 lecple 16574 glbcglb 17555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-glb 17587 |
This theorem is referenced by: glbprop 17611 meetcl 17632 clatlem 17723 op0cl 36322 atl0cl 36441 |
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