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Mirrors > Home > MPE Home > Th. List > Mathboxes > breprexplemb | Structured version Visualization version GIF version |
Description: Lemma for breprexp 34134 (closure). (Contributed by Thierry Arnoux, 7-Dec-2021.) |
Ref | Expression |
---|---|
breprexp.n | β’ (π β π β β0) |
breprexp.s | β’ (π β π β β0) |
breprexp.z | β’ (π β π β β) |
breprexp.h | β’ (π β πΏ:(0..^π)βΆ(β βm β)) |
breprexplemb.x | β’ (π β π β (0..^π)) |
breprexplemb.y | β’ (π β π β β) |
Ref | Expression |
---|---|
breprexplemb | β’ (π β ((πΏβπ)βπ) β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breprexp.h | . . . 4 β’ (π β πΏ:(0..^π)βΆ(β βm β)) | |
2 | breprexplemb.x | . . . 4 β’ (π β π β (0..^π)) | |
3 | 1, 2 | ffvelcdmd 7077 | . . 3 β’ (π β (πΏβπ) β (β βm β)) |
4 | cnex 11187 | . . . 4 β’ β β V | |
5 | nnex 12215 | . . . 4 β’ β β V | |
6 | 4, 5 | elmap 8861 | . . 3 β’ ((πΏβπ) β (β βm β) β (πΏβπ):ββΆβ) |
7 | 3, 6 | sylib 217 | . 2 β’ (π β (πΏβπ):ββΆβ) |
8 | breprexplemb.y | . 2 β’ (π β π β β) | |
9 | 7, 8 | ffvelcdmd 7077 | 1 β’ (π β ((πΏβπ)βπ) β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2098 βΆwf 6529 βcfv 6533 (class class class)co 7401 βm cmap 8816 βcc 11104 0cc0 11106 βcn 12209 β0cn0 12469 ..^cfzo 13624 |
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 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-1cn 11164 ax-addcl 11166 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-map 8818 df-nn 12210 |
This theorem is referenced by: breprexplemc 34133 circlemeth 34141 |
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