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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lubprlem | Structured version Visualization version GIF version | ||
| Description: Lemma for lubprdm 48860 and lubpr 48861. (Contributed by Zhi Wang, 26-Sep-2024.) |
| Ref | Expression |
|---|---|
| lubpr.k | ⊢ (𝜑 → 𝐾 ∈ Poset) |
| lubpr.b | ⊢ 𝐵 = (Base‘𝐾) |
| lubpr.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| lubpr.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| lubpr.l | ⊢ ≤ = (le‘𝐾) |
| lubpr.c | ⊢ (𝜑 → 𝑋 ≤ 𝑌) |
| lubpr.s | ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌}) |
| lubpr.u | ⊢ 𝑈 = (lub‘𝐾) |
| Ref | Expression |
|---|---|
| lubprlem | ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ∧ (𝑈‘𝑆) = 𝑌)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lubpr.s | . . 3 ⊢ (𝜑 → 𝑆 = {𝑋, 𝑌}) | |
| 2 | lubpr.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ Poset) | |
| 3 | breq1 5146 | . . . . . . 7 ⊢ (𝑧 = 𝑋 → (𝑧 ≤ 𝑌 ↔ 𝑋 ≤ 𝑌)) | |
| 4 | lubpr.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 5 | lubpr.c | . . . . . . 7 ⊢ (𝜑 → 𝑋 ≤ 𝑌) | |
| 6 | 3, 4, 5 | elrabd 3694 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ {𝑧 ∈ 𝐵 ∣ 𝑧 ≤ 𝑌}) |
| 7 | breq1 5146 | . . . . . . 7 ⊢ (𝑧 = 𝑌 → (𝑧 ≤ 𝑌 ↔ 𝑌 ≤ 𝑌)) | |
| 8 | lubpr.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 9 | lubpr.b | . . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾) | |
| 10 | lubpr.l | . . . . . . . . 9 ⊢ ≤ = (le‘𝐾) | |
| 11 | 9, 10 | posref 18364 | . . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ 𝑌) |
| 12 | 2, 8, 11 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ≤ 𝑌) |
| 13 | 7, 8, 12 | elrabd 3694 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ {𝑧 ∈ 𝐵 ∣ 𝑧 ≤ 𝑌}) |
| 14 | 6, 13 | prssd 4822 | . . . . 5 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ {𝑧 ∈ 𝐵 ∣ 𝑧 ≤ 𝑌}) |
| 15 | lubpr.u | . . . . 5 ⊢ 𝑈 = (lub‘𝐾) | |
| 16 | 9, 10, 15, 2, 8 | lublecl 18406 | . . . . 5 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ 𝑧 ≤ 𝑌} ∈ dom 𝑈) |
| 17 | 9, 10, 15, 2, 8 | lubid 18407 | . . . . . 6 ⊢ (𝜑 → (𝑈‘{𝑧 ∈ 𝐵 ∣ 𝑧 ≤ 𝑌}) = 𝑌) |
| 18 | prid2g 4761 | . . . . . . 7 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ {𝑋, 𝑌}) | |
| 19 | 8, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ {𝑋, 𝑌}) |
| 20 | 17, 19 | eqeltrd 2841 | . . . . 5 ⊢ (𝜑 → (𝑈‘{𝑧 ∈ 𝐵 ∣ 𝑧 ≤ 𝑌}) ∈ {𝑋, 𝑌}) |
| 21 | 2, 14, 15, 16, 20 | lubsscl 48857 | . . . 4 ⊢ (𝜑 → ({𝑋, 𝑌} ∈ dom 𝑈 ∧ (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑧 ∈ 𝐵 ∣ 𝑧 ≤ 𝑌}))) |
| 22 | 21 | simpld 494 | . . 3 ⊢ (𝜑 → {𝑋, 𝑌} ∈ dom 𝑈) |
| 23 | 1, 22 | eqeltrd 2841 | . 2 ⊢ (𝜑 → 𝑆 ∈ dom 𝑈) |
| 24 | 1 | fveq2d 6910 | . . 3 ⊢ (𝜑 → (𝑈‘𝑆) = (𝑈‘{𝑋, 𝑌})) |
| 25 | 21 | simprd 495 | . . 3 ⊢ (𝜑 → (𝑈‘{𝑋, 𝑌}) = (𝑈‘{𝑧 ∈ 𝐵 ∣ 𝑧 ≤ 𝑌})) |
| 26 | 24, 25, 17 | 3eqtrd 2781 | . 2 ⊢ (𝜑 → (𝑈‘𝑆) = 𝑌) |
| 27 | 23, 26 | jca 511 | 1 ⊢ (𝜑 → (𝑆 ∈ dom 𝑈 ∧ (𝑈‘𝑆) = 𝑌)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 {cpr 4628 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 Basecbs 17247 lecple 17304 Posetcpo 18353 lubclub 18355 |
| 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 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-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-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-id 5578 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-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-proset 18340 df-poset 18359 df-lub 18391 |
| This theorem is referenced by: lubprdm 48860 lubpr 48861 |
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