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Mirrors > Home > ILE Home > Th. List > dvmptclx | GIF version |
Description: Closure lemma for dvmptmulx 14422 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 7948 | . . . . . . 7 β’ β β V | |
3 | 2 | a1i 9 | . . . . . 6 β’ (π β β β V) |
4 | 1 | elexd 2762 | . . . . . 6 β’ (π β π β V) |
5 | dvmptadd.a | . . . . . . 7 β’ ((π β§ π₯ β π) β π΄ β β) | |
6 | 5 | fmpttd 5684 | . . . . . 6 β’ (π β (π₯ β π β¦ π΄):πβΆβ) |
7 | dvmptclx.ss | . . . . . 6 β’ (π β π β π) | |
8 | elpm2r 6679 | . . . . . 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 1249 | . . . . 5 β’ (π β (π₯ β π β¦ π΄) β (β βpm π)) |
10 | dvfgg 14397 | . . . . 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 4841 | . . . . . 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 2560 | . . . . . . 7 β’ (π β βπ₯ β π π΅ β π) |
16 | dmmptg 5138 | . . . . . . 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 2220 | . . . . 5 β’ (π β dom (π D (π₯ β π β¦ π΄)) = π) |
19 | 18 | feq2d 5365 | . . . 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 5364 | . . 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 5682 | 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 1363 β wcel 2158 βwral 2465 Vcvv 2749 β wss 3141 {cpr 3605 β¦ cmpt 4076 dom cdm 4638 βΆwf 5224 (class class class)co 5888 βpm cpm 6662 βcc 7822 βcr 7823 D cdv 14364 |
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-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 ax-arch 7943 ax-caucvg 7944 |
This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 980 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-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 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-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-frec 6405 df-map 6663 df-pm 6664 df-sup 6996 df-inf 6997 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-n0 9190 df-z 9267 df-uz 9542 df-q 9633 df-rp 9667 df-xneg 9785 df-xadd 9786 df-seqfrec 10459 df-exp 10533 df-cj 10864 df-re 10865 df-im 10866 df-rsqrt 11020 df-abs 11021 df-rest 12707 df-topgen 12726 df-psmet 13673 df-xmet 13674 df-met 13675 df-bl 13676 df-mopn 13677 df-top 13738 df-topon 13751 df-bases 13783 df-ntr 13836 df-limced 14365 df-dvap 14366 |
This theorem is referenced by: dvmptmulx 14422 dvmptcmulcn 14423 dvmptnegcn 14424 dvmptsubcn 14425 dvmptcjx 14426 |
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