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Mirrors > Home > ILE Home > Th. List > dvlemap | GIF version |
Description: Closure for a difference quotient. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 27-Jun-2023.) |
Ref | Expression |
---|---|
dvlem.1 | β’ (π β πΉ:π·βΆβ) |
dvlem.2 | β’ (π β π· β β) |
dvlem.3 | β’ (π β π΅ β π·) |
Ref | Expression |
---|---|
dvlemap | β’ ((π β§ π΄ β {π€ β π· β£ π€ # π΅}) β (((πΉβπ΄) β (πΉβπ΅)) / (π΄ β π΅)) β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvlem.1 | . . . . 5 β’ (π β πΉ:π·βΆβ) | |
2 | 1 | adantr 276 | . . . 4 β’ ((π β§ π΄ β {π€ β π· β£ π€ # π΅}) β πΉ:π·βΆβ) |
3 | elrabi 2892 | . . . . 5 β’ (π΄ β {π€ β π· β£ π€ # π΅} β π΄ β π·) | |
4 | 3 | adantl 277 | . . . 4 β’ ((π β§ π΄ β {π€ β π· β£ π€ # π΅}) β π΄ β π·) |
5 | 2, 4 | ffvelcdmd 5654 | . . 3 β’ ((π β§ π΄ β {π€ β π· β£ π€ # π΅}) β (πΉβπ΄) β β) |
6 | dvlem.3 | . . . . 5 β’ (π β π΅ β π·) | |
7 | 6 | adantr 276 | . . . 4 β’ ((π β§ π΄ β {π€ β π· β£ π€ # π΅}) β π΅ β π·) |
8 | 2, 7 | ffvelcdmd 5654 | . . 3 β’ ((π β§ π΄ β {π€ β π· β£ π€ # π΅}) β (πΉβπ΅) β β) |
9 | 5, 8 | subcld 8270 | . 2 β’ ((π β§ π΄ β {π€ β π· β£ π€ # π΅}) β ((πΉβπ΄) β (πΉβπ΅)) β β) |
10 | dvlem.2 | . . . . 5 β’ (π β π· β β) | |
11 | 10 | adantr 276 | . . . 4 β’ ((π β§ π΄ β {π€ β π· β£ π€ # π΅}) β π· β β) |
12 | 11, 4 | sseldd 3158 | . . 3 β’ ((π β§ π΄ β {π€ β π· β£ π€ # π΅}) β π΄ β β) |
13 | 10, 6 | sseldd 3158 | . . . 4 β’ (π β π΅ β β) |
14 | 13 | adantr 276 | . . 3 β’ ((π β§ π΄ β {π€ β π· β£ π€ # π΅}) β π΅ β β) |
15 | 12, 14 | subcld 8270 | . 2 β’ ((π β§ π΄ β {π€ β π· β£ π€ # π΅}) β (π΄ β π΅) β β) |
16 | breq1 4008 | . . . . . 6 β’ (π€ = π΄ β (π€ # π΅ β π΄ # π΅)) | |
17 | 16 | elrab 2895 | . . . . 5 β’ (π΄ β {π€ β π· β£ π€ # π΅} β (π΄ β π· β§ π΄ # π΅)) |
18 | 17 | simprbi 275 | . . . 4 β’ (π΄ β {π€ β π· β£ π€ # π΅} β π΄ # π΅) |
19 | 18 | adantl 277 | . . 3 β’ ((π β§ π΄ β {π€ β π· β£ π€ # π΅}) β π΄ # π΅) |
20 | 12, 14, 19 | subap0d 8603 | . 2 β’ ((π β§ π΄ β {π€ β π· β£ π€ # π΅}) β (π΄ β π΅) # 0) |
21 | 9, 15, 20 | divclapd 8749 | 1 β’ ((π β§ π΄ β {π€ β π· β£ π€ # π΅}) β (((πΉβπ΄) β (πΉβπ΅)) / (π΄ β π΅)) β β) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 β wcel 2148 {crab 2459 β wss 3131 class class class wbr 4005 βΆwf 5214 βcfv 5218 (class class class)co 5877 βcc 7811 β cmin 8130 # cap 8540 / cdiv 8631 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-id 4295 df-po 4298 df-iso 4299 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 |
This theorem is referenced by: dvfgg 14242 dvcnp2cntop 14248 dvaddxxbr 14250 dvmulxxbr 14251 dvcoapbr 14256 dvcjbr 14257 |
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