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Mirrors > Home > ILE Home > Th. List > dvmptclx | GIF version |
Description: Closure lemma for dvmptmulx 14585 and other related theorems. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
dvmptadd.s | β’ (π β π β {β, β}) |
dvmptadd.a | β’ ((π β§ π₯ β π) β π΄ β β) |
dvmptadd.b | β’ ((π β§ π₯ β π) β π΅ β π) |
dvmptadd.da | β’ (π β (π D (π₯ β π β¦ π΄)) = (π₯ β π β¦ π΅)) |
dvmptclx.ss | β’ (π β π β π) |
Ref | Expression |
---|---|
dvmptclx | β’ ((π β§ π₯ β π) β π΅ β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvmptadd.s | . . . . 5 β’ (π β π β {β, β}) | |
2 | cnex 7954 | . . . . . . 7 β’ β β V | |
3 | 2 | a1i 9 | . . . . . 6 β’ (π β β β V) |
4 | 1 | elexd 2765 | . . . . . 6 β’ (π β π β V) |
5 | dvmptadd.a | . . . . . . 7 β’ ((π β§ π₯ β π) β π΄ β β) | |
6 | 5 | fmpttd 5687 | . . . . . 6 β’ (π β (π₯ β π β¦ π΄):πβΆβ) |
7 | dvmptclx.ss | . . . . . 6 β’ (π β π β π) | |
8 | elpm2r 6684 | . . . . . 6 β’ (((β β V β§ π β V) β§ ((π₯ β π β¦ π΄):πβΆβ β§ π β π)) β (π₯ β π β¦ π΄) β (β βpm π)) | |
9 | 3, 4, 6, 7, 8 | syl22anc 1250 | . . . . 5 β’ (π β (π₯ β π β¦ π΄) β (β βpm π)) |
10 | dvfgg 14560 | . . . . 5 β’ ((π β {β, β} β§ (π₯ β π β¦ π΄) β (β βpm π)) β (π D (π₯ β π β¦ π΄)):dom (π D (π₯ β π β¦ π΄))βΆβ) | |
11 | 1, 9, 10 | syl2anc 411 | . . . 4 β’ (π β (π D (π₯ β π β¦ π΄)):dom (π D (π₯ β π β¦ π΄))βΆβ) |
12 | dvmptadd.da | . . . . . . 7 β’ (π β (π D (π₯ β π β¦ π΄)) = (π₯ β π β¦ π΅)) | |
13 | 12 | dmeqd 4844 | . . . . . 6 β’ (π β dom (π D (π₯ β π β¦ π΄)) = dom (π₯ β π β¦ π΅)) |
14 | dvmptadd.b | . . . . . . . 8 β’ ((π β§ π₯ β π) β π΅ β π) | |
15 | 14 | ralrimiva 2563 | . . . . . . 7 β’ (π β βπ₯ β π π΅ β π) |
16 | dmmptg 5141 | . . . . . . 7 β’ (βπ₯ β π π΅ β π β dom (π₯ β π β¦ π΅) = π) | |
17 | 15, 16 | syl 14 | . . . . . 6 β’ (π β dom (π₯ β π β¦ π΅) = π) |
18 | 13, 17 | eqtrd 2222 | . . . . 5 β’ (π β dom (π D (π₯ β π β¦ π΄)) = π) |
19 | 18 | feq2d 5368 | . . . 4 β’ (π β ((π D (π₯ β π β¦ π΄)):dom (π D (π₯ β π β¦ π΄))βΆβ β (π D (π₯ β π β¦ π΄)):πβΆβ)) |
20 | 11, 19 | mpbid 147 | . . 3 β’ (π β (π D (π₯ β π β¦ π΄)):πβΆβ) |
21 | 12 | feq1d 5367 | . . 3 β’ (π β ((π D (π₯ β π β¦ π΄)):πβΆβ β (π₯ β π β¦ π΅):πβΆβ)) |
22 | 20, 21 | mpbid 147 | . 2 β’ (π β (π₯ β π β¦ π΅):πβΆβ) |
23 | 22 | fvmptelcdm 5685 | 1 β’ ((π β§ π₯ β π) β π΅ β β) |
Colors of variables: wff set class |
Syntax hints: β wi 4 β§ wa 104 = wceq 1364 β wcel 2160 βwral 2468 Vcvv 2752 β wss 3144 {cpr 3608 β¦ cmpt 4079 dom cdm 4641 βΆwf 5227 (class class class)co 5891 βpm cpm 6667 βcc 7828 βcr 7829 D cdv 14527 |
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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-mulrcl 7929 ax-addcom 7930 ax-mulcom 7931 ax-addass 7932 ax-mulass 7933 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-1rid 7937 ax-0id 7938 ax-rnegex 7939 ax-precex 7940 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-apti 7945 ax-pre-ltadd 7946 ax-pre-mulgt0 7947 ax-pre-mulext 7948 ax-arch 7949 ax-caucvg 7950 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-isom 5240 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-frec 6410 df-map 6668 df-pm 6669 df-sup 7002 df-inf 7003 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-reap 8551 df-ap 8558 df-div 8649 df-inn 8939 df-2 8997 df-3 8998 df-4 8999 df-n0 9196 df-z 9273 df-uz 9548 df-q 9639 df-rp 9673 df-xneg 9791 df-xadd 9792 df-seqfrec 10465 df-exp 10539 df-cj 10870 df-re 10871 df-im 10872 df-rsqrt 11026 df-abs 11027 df-rest 12718 df-topgen 12737 df-psmet 13823 df-xmet 13824 df-met 13825 df-bl 13826 df-mopn 13827 df-top 13901 df-topon 13914 df-bases 13946 df-ntr 13999 df-limced 14528 df-dvap 14529 |
This theorem is referenced by: dvmptmulx 14585 dvmptcmulcn 14586 dvmptnegcn 14587 dvmptsubcn 14588 dvmptcjx 14589 |
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