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Mirrors > Home > MPE Home > Th. List > htpyco1 | Structured version Visualization version Unicode version |
Description: Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.) |
Ref | Expression |
---|---|
htpyco1.n |
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htpyco1.j |
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htpyco1.p |
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htpyco1.f |
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htpyco1.g |
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htpyco1.h |
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Ref | Expression |
---|---|
htpyco1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | htpyco1.j |
. 2
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2 | htpyco1.p |
. . 3
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3 | htpyco1.f |
. . 3
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4 | cnco 20359 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 2, 3, 4 | syl2anc 673 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | htpyco1.g |
. . 3
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7 | cnco 20359 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 2, 6, 7 | syl2anc 673 |
. 2
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9 | htpyco1.n |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | iitopon 21989 |
. . . . 5
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11 | 10 | a1i 11 |
. . . 4
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12 | 1, 11 | cnmpt1st 20760 |
. . . . 5
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13 | 1, 11, 12, 2 | cnmpt21f 20764 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 1, 11 | cnmpt2nd 20761 |
. . . 4
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15 | cntop2 20334 |
. . . . . . . 8
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16 | 2, 15 | syl 17 |
. . . . . . 7
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17 | eqid 2471 |
. . . . . . . 8
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18 | 17 | toptopon 20025 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 16, 18 | sylib 201 |
. . . . . 6
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20 | 19, 3, 6 | htpycn 22082 |
. . . . 5
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21 | htpyco1.h |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
22 | 20, 21 | sseldd 3419 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 1, 11, 13, 14, 22 | cnmpt22f 20767 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 9, 23 | syl5eqel 2553 |
. 2
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25 | cnf2 20342 |
. . . . . . 7
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26 | 1, 19, 2, 25 | syl3anc 1292 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 26 | ffvelrnda 6037 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 19, 3, 6, 21 | htpyi 22083 |
. . . . 5
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29 | 27, 28 | syldan 478 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
30 | 29 | simpld 466 |
. . 3
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31 | simpr 468 |
. . . 4
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32 | 0elunit 11776 |
. . . 4
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33 | fveq2 5879 |
. . . . . 6
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34 | id 22 |
. . . . . 6
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35 | 33, 34 | oveqan12d 6327 |
. . . . 5
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36 | ovex 6336 |
. . . . 5
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37 | 35, 9, 36 | ovmpt2a 6446 |
. . . 4
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38 | 31, 32, 37 | sylancl 675 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | fvco3 5957 |
. . . 4
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40 | 26, 39 | sylan 479 |
. . 3
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41 | 30, 38, 40 | 3eqtr4d 2515 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
42 | 29 | simprd 470 |
. . 3
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43 | 1elunit 11777 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
44 | id 22 |
. . . . . 6
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45 | 33, 44 | oveqan12d 6327 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
46 | ovex 6336 |
. . . . 5
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47 | 45, 9, 46 | ovmpt2a 6446 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | 31, 43, 47 | sylancl 675 |
. . 3
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49 | fvco3 5957 |
. . . 4
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50 | 26, 49 | sylan 479 |
. . 3
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51 | 42, 48, 50 | 3eqtr4d 2515 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
52 | 1, 5, 8, 24, 41, 51 | ishtpyd 22084 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 ax-cnex 9613 ax-resscn 9614 ax-1cn 9615 ax-icn 9616 ax-addcl 9617 ax-addrcl 9618 ax-mulcl 9619 ax-mulrcl 9620 ax-mulcom 9621 ax-addass 9622 ax-mulass 9623 ax-distr 9624 ax-i2m1 9625 ax-1ne0 9626 ax-1rid 9627 ax-rnegex 9628 ax-rrecex 9629 ax-cnre 9630 ax-pre-lttri 9631 ax-pre-lttrn 9632 ax-pre-ltadd 9633 ax-pre-mulgt0 9634 ax-pre-sup 9635 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3or 1008 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-nel 2644 df-ral 2761 df-rex 2762 df-reu 2763 df-rmo 2764 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-pss 3406 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-tp 3964 df-op 3966 df-uni 4191 df-iun 4271 df-br 4396 df-opab 4455 df-mpt 4456 df-tr 4491 df-eprel 4750 df-id 4754 df-po 4760 df-so 4761 df-fr 4798 df-we 4800 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-pred 5387 df-ord 5433 df-on 5434 df-lim 5435 df-suc 5436 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-riota 6270 df-ov 6311 df-oprab 6312 df-mpt2 6313 df-om 6712 df-1st 6812 df-2nd 6813 df-wrecs 7046 df-recs 7108 df-rdg 7146 df-er 7381 df-map 7492 df-en 7588 df-dom 7589 df-sdom 7590 df-sup 7974 df-inf 7975 df-pnf 9695 df-mnf 9696 df-xr 9697 df-ltxr 9698 df-le 9699 df-sub 9882 df-neg 9883 df-div 10292 df-nn 10632 df-2 10690 df-3 10691 df-n0 10894 df-z 10962 df-uz 11183 df-q 11288 df-rp 11326 df-xneg 11432 df-xadd 11433 df-xmul 11434 df-icc 11667 df-seq 12252 df-exp 12311 df-cj 13239 df-re 13240 df-im 13241 df-sqrt 13375 df-abs 13376 df-topgen 15420 df-psmet 19039 df-xmet 19040 df-met 19041 df-bl 19042 df-mopn 19043 df-top 19998 df-bases 19999 df-topon 20000 df-cn 20320 df-tx 20654 df-ii 21987 df-htpy 22079 |
This theorem is referenced by: (None) |
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