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Mirrors > Home > MPE Home > Th. List > infiso | Structured version Visualization version GIF version |
Description: Image of an infimum under an isomorphism. (Contributed by AV, 4-Sep-2020.) |
Ref | Expression |
---|---|
infiso.1 | β’ (π β πΉ Isom π , π (π΄, π΅)) |
infiso.2 | β’ (π β πΆ β π΄) |
infiso.3 | β’ (π β βπ₯ β π΄ (βπ¦ β πΆ Β¬ π¦π π₯ β§ βπ¦ β π΄ (π₯π π¦ β βπ§ β πΆ π§π π¦))) |
infiso.4 | β’ (π β π Or π΄) |
Ref | Expression |
---|---|
infiso | β’ (π β inf((πΉ β πΆ), π΅, π) = (πΉβinf(πΆ, π΄, π ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infiso.1 | . . . 4 β’ (π β πΉ Isom π , π (π΄, π΅)) | |
2 | isocnv2 7323 | . . . 4 β’ (πΉ Isom π , π (π΄, π΅) β πΉ Isom β‘π , β‘π(π΄, π΅)) | |
3 | 1, 2 | sylib 217 | . . 3 β’ (π β πΉ Isom β‘π , β‘π(π΄, π΅)) |
4 | infiso.2 | . . 3 β’ (π β πΆ β π΄) | |
5 | infiso.4 | . . . 4 β’ (π β π Or π΄) | |
6 | infiso.3 | . . . 4 β’ (π β βπ₯ β π΄ (βπ¦ β πΆ Β¬ π¦π π₯ β§ βπ¦ β π΄ (π₯π π¦ β βπ§ β πΆ π§π π¦))) | |
7 | 5, 6 | infcllem 9481 | . . 3 β’ (π β βπ₯ β π΄ (βπ¦ β πΆ Β¬ π₯β‘π π¦ β§ βπ¦ β π΄ (π¦β‘π π₯ β βπ§ β πΆ π¦β‘π π§))) |
8 | cnvso 6280 | . . . 4 β’ (π Or π΄ β β‘π Or π΄) | |
9 | 5, 8 | sylib 217 | . . 3 β’ (π β β‘π Or π΄) |
10 | 3, 4, 7, 9 | supiso 9469 | . 2 β’ (π β sup((πΉ β πΆ), π΅, β‘π) = (πΉβsup(πΆ, π΄, β‘π ))) |
11 | df-inf 9437 | . 2 β’ inf((πΉ β πΆ), π΅, π) = sup((πΉ β πΆ), π΅, β‘π) | |
12 | df-inf 9437 | . . 3 β’ inf(πΆ, π΄, π ) = sup(πΆ, π΄, β‘π ) | |
13 | 12 | fveq2i 6887 | . 2 β’ (πΉβinf(πΆ, π΄, π )) = (πΉβsup(πΆ, π΄, β‘π )) |
14 | 10, 11, 13 | 3eqtr4g 2791 | 1 β’ (π β inf((πΉ β πΆ), π΅, π) = (πΉβinf(πΆ, π΄, π ))) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 = wceq 1533 βwral 3055 βwrex 3064 β wss 3943 class class class wbr 5141 Or wor 5580 β‘ccnv 5668 β cima 5672 βcfv 6536 Isom wiso 6537 supcsup 9434 infcinf 9435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-sup 9436 df-inf 9437 |
This theorem is referenced by: (None) |
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