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Mirrors > Home > ILE Home > Th. List > infisoti | GIF version |
Description: Image of an infimum under an isomorphism. (Contributed by Jim Kingdon, 19-Dec-2021.) |
Ref | Expression |
---|---|
infisoti.1 | β’ (π β πΉ Isom π , π (π΄, π΅)) |
infisoti.2 | β’ (π β πΆ β π΄) |
infisoti.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¦))) |
infisoti.ti | β’ ((π β§ (π’ β π΄ β§ π£ β π΄)) β (π’ = π£ β (Β¬ π’π π£ β§ Β¬ π£π π’))) |
Ref | Expression |
---|---|
infisoti | β’ (π β inf((πΉ β πΆ), π΅, π) = (πΉβinf(πΆ, π΄, π ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infisoti.1 | . . . 4 β’ (π β πΉ Isom π , π (π΄, π΅)) | |
2 | isocnv2 5826 | . . . 4 β’ (πΉ Isom π , π (π΄, π΅) β πΉ Isom β‘π , β‘π(π΄, π΅)) | |
3 | 1, 2 | sylib 122 | . . 3 β’ (π β πΉ Isom β‘π , β‘π(π΄, π΅)) |
4 | infisoti.2 | . . 3 β’ (π β πΆ β π΄) | |
5 | infisoti.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¦))) | |
6 | 5 | cnvinfex 7030 | . . 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§))) |
7 | infisoti.ti | . . . 4 β’ ((π β§ (π’ β π΄ β§ π£ β π΄)) β (π’ = π£ β (Β¬ π’π π£ β§ Β¬ π£π π’))) | |
8 | 7 | cnvti 7031 | . . 3 β’ ((π β§ (π’ β π΄ β§ π£ β π΄)) β (π’ = π£ β (Β¬ π’β‘π π£ β§ Β¬ π£β‘π π’))) |
9 | 3, 4, 6, 8 | supisoti 7022 | . 2 β’ (π β sup((πΉ β πΆ), π΅, β‘π) = (πΉβsup(πΆ, π΄, β‘π ))) |
10 | df-inf 6997 | . 2 β’ inf((πΉ β πΆ), π΅, π) = sup((πΉ β πΆ), π΅, β‘π) | |
11 | df-inf 6997 | . . 3 β’ inf(πΆ, π΄, π ) = sup(πΆ, π΄, β‘π ) | |
12 | 11 | fveq2i 5530 | . 2 β’ (πΉβinf(πΆ, π΄, π )) = (πΉβsup(πΆ, π΄, β‘π )) |
13 | 9, 10, 12 | 3eqtr4g 2245 | 1 β’ (π β inf((πΉ β πΆ), π΅, π) = (πΉβinf(πΆ, π΄, π ))) |
Colors of variables: wff set class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 104 β wb 105 = wceq 1363 β wcel 2158 βwral 2465 βwrex 2466 β wss 3141 class class class wbr 4015 β‘ccnv 4637 β cima 4641 βcfv 5228 Isom wiso 5229 supcsup 6994 infcinf 6995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-isom 5237 df-riota 5844 df-sup 6996 df-inf 6997 |
This theorem is referenced by: (None) |
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