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Mirrors > Home > MPE Home > Th. List > meetval2lem | Structured version Visualization version GIF version |
Description: Lemma for meetval2 18372 and meeteu 18373. (Contributed by NM, 12-Sep-2018.) TODO: combine this through meeteu 18373 into meetlem 18374? |
Ref | Expression |
---|---|
meetval2.b | β’ π΅ = (BaseβπΎ) |
meetval2.l | β’ β€ = (leβπΎ) |
meetval2.m | β’ β§ = (meetβπΎ) |
meetval2.k | β’ (π β πΎ β π) |
meetval2.x | β’ (π β π β π΅) |
meetval2.y | β’ (π β π β π΅) |
Ref | Expression |
---|---|
meetval2lem | β’ ((π β π΅ β§ π β π΅) β ((βπ¦ β {π, π}π₯ β€ π¦ β§ βπ§ β π΅ (βπ¦ β {π, π}π§ β€ π¦ β π§ β€ π₯)) β ((π₯ β€ π β§ π₯ β€ π) β§ βπ§ β π΅ ((π§ β€ π β§ π§ β€ π) β π§ β€ π₯)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 5146 | . . 3 β’ (π¦ = π β (π₯ β€ π¦ β π₯ β€ π)) | |
2 | breq2 5146 | . . 3 β’ (π¦ = π β (π₯ β€ π¦ β π₯ β€ π)) | |
3 | 1, 2 | ralprg 4694 | . 2 β’ ((π β π΅ β§ π β π΅) β (βπ¦ β {π, π}π₯ β€ π¦ β (π₯ β€ π β§ π₯ β€ π))) |
4 | breq2 5146 | . . . . 5 β’ (π¦ = π β (π§ β€ π¦ β π§ β€ π)) | |
5 | breq2 5146 | . . . . 5 β’ (π¦ = π β (π§ β€ π¦ β π§ β€ π)) | |
6 | 4, 5 | ralprg 4694 | . . . 4 β’ ((π β π΅ β§ π β π΅) β (βπ¦ β {π, π}π§ β€ π¦ β (π§ β€ π β§ π§ β€ π))) |
7 | 6 | imbi1d 341 | . . 3 β’ ((π β π΅ β§ π β π΅) β ((βπ¦ β {π, π}π§ β€ π¦ β π§ β€ π₯) β ((π§ β€ π β§ π§ β€ π) β π§ β€ π₯))) |
8 | 7 | ralbidv 3172 | . 2 β’ ((π β π΅ β§ π β π΅) β (βπ§ β π΅ (βπ¦ β {π, π}π§ β€ π¦ β π§ β€ π₯) β βπ§ β π΅ ((π§ β€ π β§ π§ β€ π) β π§ β€ π₯))) |
9 | 3, 8 | anbi12d 630 | 1 β’ ((π β π΅ β§ π β π΅) β ((βπ¦ β {π, π}π₯ β€ π¦ β§ βπ§ β π΅ (βπ¦ β {π, π}π§ β€ π¦ β π§ β€ π₯)) β ((π₯ β€ π β§ π₯ β€ π) β§ βπ§ β π΅ ((π§ β€ π β§ π§ β€ π) β π§ β€ π₯)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1534 β wcel 2099 βwral 3056 {cpr 4626 class class class wbr 5142 βcfv 6542 Basecbs 17165 lecple 17225 meetcmee 18289 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 |
This theorem is referenced by: meetval2 18372 meeteu 18373 meetdm3 47903 |
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