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Mirrors > Home > MPE Home > Th. List > Mathboxes > lubprlem | Structured version Visualization version GIF version |
Description: Lemma for lubprdm 48760 and lubpr 48761. (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 5151 | . . . . . . 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 3697 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ {𝑧 ∈ 𝐵 ∣ 𝑧 ≤ 𝑌}) |
7 | breq1 5151 | . . . . . . 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 18376 | . . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ 𝑌) |
12 | 2, 8, 11 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ≤ 𝑌) |
13 | 7, 8, 12 | elrabd 3697 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ {𝑧 ∈ 𝐵 ∣ 𝑧 ≤ 𝑌}) |
14 | 6, 13 | prssd 4827 | . . . . 5 ⊢ (𝜑 → {𝑋, 𝑌} ⊆ {𝑧 ∈ 𝐵 ∣ 𝑧 ≤ 𝑌}) |
15 | lubpr.u | . . . . 5 ⊢ 𝑈 = (lub‘𝐾) | |
16 | 9, 10, 15, 2, 8 | lublecl 18419 | . . . . 5 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ 𝑧 ≤ 𝑌} ∈ dom 𝑈) |
17 | 9, 10, 15, 2, 8 | lubid 18420 | . . . . . 6 ⊢ (𝜑 → (𝑈‘{𝑧 ∈ 𝐵 ∣ 𝑧 ≤ 𝑌}) = 𝑌) |
18 | prid2g 4766 | . . . . . . 7 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ {𝑋, 𝑌}) | |
19 | 8, 18 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ {𝑋, 𝑌}) |
20 | 17, 19 | eqeltrd 2839 | . . . . 5 ⊢ (𝜑 → (𝑈‘{𝑧 ∈ 𝐵 ∣ 𝑧 ≤ 𝑌}) ∈ {𝑋, 𝑌}) |
21 | 2, 14, 15, 16, 20 | lubsscl 48757 | . . . 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 2839 | . 2 ⊢ (𝜑 → 𝑆 ∈ dom 𝑈) |
24 | 1 | fveq2d 6911 | . . 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 2779 | . 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 1537 ∈ wcel 2106 {crab 3433 {cpr 4633 class class class wbr 5148 dom cdm 5689 ‘cfv 6563 Basecbs 17245 lecple 17305 Posetcpo 18365 lubclub 18367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-proset 18352 df-poset 18371 df-lub 18404 |
This theorem is referenced by: lubprdm 48760 lubpr 48761 |
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