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Mirrors > Home > MPE Home > Th. List > coe1mul2lem1 | Structured version Visualization version GIF version |
Description: An equivalence for coe1mul2 21656. (Contributed by Stefan O'Rear, 25-Mar-2015.) |
Ref | Expression |
---|---|
coe1mul2lem1 | β’ ((π΄ β β0 β§ π β (β0 βm 1o)) β (π βr β€ (1o Γ {π΄}) β (πββ ) β (0...π΄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 8425 | . . . 4 β’ 1o β On | |
2 | 1 | a1i 11 | . . 3 β’ ((π΄ β β0 β§ π β (β0 βm 1o)) β 1o β On) |
3 | fvexd 6858 | . . 3 β’ (((π΄ β β0 β§ π β (β0 βm 1o)) β§ π β 1o) β (πββ ) β V) | |
4 | simpll 766 | . . 3 β’ (((π΄ β β0 β§ π β (β0 βm 1o)) β§ π β 1o) β π΄ β β0) | |
5 | df1o2 8420 | . . . . . 6 β’ 1o = {β } | |
6 | nn0ex 12424 | . . . . . 6 β’ β0 β V | |
7 | 0ex 5265 | . . . . . 6 β’ β β V | |
8 | 5, 6, 7 | mapsnconst 8833 | . . . . 5 β’ (π β (β0 βm 1o) β π = (1o Γ {(πββ )})) |
9 | 8 | adantl 483 | . . . 4 β’ ((π΄ β β0 β§ π β (β0 βm 1o)) β π = (1o Γ {(πββ )})) |
10 | fconstmpt 5695 | . . . 4 β’ (1o Γ {(πββ )}) = (π β 1o β¦ (πββ )) | |
11 | 9, 10 | eqtrdi 2789 | . . 3 β’ ((π΄ β β0 β§ π β (β0 βm 1o)) β π = (π β 1o β¦ (πββ ))) |
12 | fconstmpt 5695 | . . . 4 β’ (1o Γ {π΄}) = (π β 1o β¦ π΄) | |
13 | 12 | a1i 11 | . . 3 β’ ((π΄ β β0 β§ π β (β0 βm 1o)) β (1o Γ {π΄}) = (π β 1o β¦ π΄)) |
14 | 2, 3, 4, 11, 13 | ofrfval2 7639 | . 2 β’ ((π΄ β β0 β§ π β (β0 βm 1o)) β (π βr β€ (1o Γ {π΄}) β βπ β 1o (πββ ) β€ π΄)) |
15 | 1n0 8435 | . . 3 β’ 1o β β | |
16 | r19.3rzv 4457 | . . 3 β’ (1o β β β ((πββ ) β€ π΄ β βπ β 1o (πββ ) β€ π΄)) | |
17 | 15, 16 | mp1i 13 | . 2 β’ ((π΄ β β0 β§ π β (β0 βm 1o)) β ((πββ ) β€ π΄ β βπ β 1o (πββ ) β€ π΄)) |
18 | elmapi 8790 | . . . . . 6 β’ (π β (β0 βm 1o) β π:1oβΆβ0) | |
19 | 0lt1o 8451 | . . . . . 6 β’ β β 1o | |
20 | ffvelcdm 7033 | . . . . . 6 β’ ((π:1oβΆβ0 β§ β β 1o) β (πββ ) β β0) | |
21 | 18, 19, 20 | sylancl 587 | . . . . 5 β’ (π β (β0 βm 1o) β (πββ ) β β0) |
22 | 21 | adantl 483 | . . . 4 β’ ((π΄ β β0 β§ π β (β0 βm 1o)) β (πββ ) β β0) |
23 | 22 | biantrurd 534 | . . 3 β’ ((π΄ β β0 β§ π β (β0 βm 1o)) β ((πββ ) β€ π΄ β ((πββ ) β β0 β§ (πββ ) β€ π΄))) |
24 | fznn0 13539 | . . . 4 β’ (π΄ β β0 β ((πββ ) β (0...π΄) β ((πββ ) β β0 β§ (πββ ) β€ π΄))) | |
25 | 24 | adantr 482 | . . 3 β’ ((π΄ β β0 β§ π β (β0 βm 1o)) β ((πββ ) β (0...π΄) β ((πββ ) β β0 β§ (πββ ) β€ π΄))) |
26 | 23, 25 | bitr4d 282 | . 2 β’ ((π΄ β β0 β§ π β (β0 βm 1o)) β ((πββ ) β€ π΄ β (πββ ) β (0...π΄))) |
27 | 14, 17, 26 | 3bitr2d 307 | 1 β’ ((π΄ β β0 β§ π β (β0 βm 1o)) β (π βr β€ (1o Γ {π΄}) β (πββ ) β (0...π΄))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2940 βwral 3061 Vcvv 3444 β c0 4283 {csn 4587 class class class wbr 5106 β¦ cmpt 5189 Γ cxp 5632 Oncon0 6318 βΆwf 6493 βcfv 6497 (class class class)co 7358 βr cofr 7617 1oc1o 8406 βm cmap 8768 0cc0 11056 β€ cle 11195 β0cn0 12418 ...cfz 13430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-ofr 7619 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 df-z 12505 df-fz 13431 |
This theorem is referenced by: coe1mul2lem2 21655 coe1mul2 21656 |
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