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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pconpi1 | Structured version Visualization version Unicode version |
Description: All fundamental groups in a path-connected space are isomorphic. (Contributed by Mario Carneiro, 12-Feb-2015.) |
Ref | Expression |
---|---|
pconpi1.x |
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pconpi1.p |
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pconpi1.q |
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pconpi1.s |
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pconpi1.t |
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Ref | Expression |
---|---|
pconpi1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pconpi1.x |
. . 3
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2 | 1 | pconcn 29940 |
. 2
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3 | eqid 2450 |
. . . . 5
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4 | eqid 2450 |
. . . . 5
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5 | eqid 2450 |
. . . . 5
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6 | eqid 2450 |
. . . . 5
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7 | simpl1 1010 |
. . . . . . 7
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8 | pcontop 29941 |
. . . . . . 7
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9 | 7, 8 | syl 17 |
. . . . . 6
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10 | 1 | toptopon 19941 |
. . . . . 6
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11 | 9, 10 | sylib 200 |
. . . . 5
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12 | simprl 763 |
. . . . 5
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13 | oveq2 6296 |
. . . . . . 7
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14 | 13 | fveq2d 5867 |
. . . . . 6
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15 | 14 | cbvmptv 4494 |
. . . . 5
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16 | 3, 4, 5, 6, 11, 12, 15 | pi1xfrgim 22082 |
. . . 4
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17 | simprrl 773 |
. . . . . . 7
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18 | 17 | oveq2d 6304 |
. . . . . 6
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19 | pconpi1.p |
. . . . . 6
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20 | 18, 19 | syl6eqr 2502 |
. . . . 5
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21 | simprrr 774 |
. . . . . . 7
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22 | 21 | oveq2d 6304 |
. . . . . 6
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23 | pconpi1.q |
. . . . . 6
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24 | 22, 23 | syl6eqr 2502 |
. . . . 5
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25 | 20, 24 | oveq12d 6306 |
. . . 4
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26 | 16, 25 | eleqtrd 2530 |
. . 3
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27 | brgici 16927 |
. . 3
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28 | 26, 27 | syl 17 |
. 2
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29 | 2, 28 | rexlimddv 2882 |
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 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-rep 4514 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 ax-inf2 8143 ax-cnex 9592 ax-resscn 9593 ax-1cn 9594 ax-icn 9595 ax-addcl 9596 ax-addrcl 9597 ax-mulcl 9598 ax-mulrcl 9599 ax-mulcom 9600 ax-addass 9601 ax-mulass 9602 ax-distr 9603 ax-i2m1 9604 ax-1ne0 9605 ax-1rid 9606 ax-rnegex 9607 ax-rrecex 9608 ax-cnre 9609 ax-pre-lttri 9610 ax-pre-lttrn 9611 ax-pre-ltadd 9612 ax-pre-mulgt0 9613 ax-pre-sup 9614 ax-addf 9615 ax-mulf 9616 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 985 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-nel 2624 df-ral 2741 df-rex 2742 df-reu 2743 df-rmo 2744 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-pss 3419 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-tp 3972 df-op 3974 df-uni 4198 df-int 4234 df-iun 4279 df-iin 4280 df-br 4402 df-opab 4461 df-mpt 4462 df-tr 4497 df-eprel 4744 df-id 4748 df-po 4754 df-so 4755 df-fr 4792 df-se 4793 df-we 4794 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-pred 5379 df-ord 5425 df-on 5426 df-lim 5427 df-suc 5428 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-isom 5590 df-riota 6250 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-of 6528 df-om 6690 df-1st 6790 df-2nd 6791 df-supp 6912 df-wrecs 7025 df-recs 7087 df-rdg 7125 df-1o 7179 df-2o 7180 df-oadd 7183 df-er 7360 df-ec 7362 df-qs 7366 df-map 7471 df-ixp 7520 df-en 7567 df-dom 7568 df-sdom 7569 df-fin 7570 df-fsupp 7881 df-fi 7922 df-sup 7953 df-inf 7954 df-oi 8022 df-card 8370 df-cda 8595 df-pnf 9674 df-mnf 9675 df-xr 9676 df-ltxr 9677 df-le 9678 df-sub 9859 df-neg 9860 df-div 10267 df-nn 10607 df-2 10665 df-3 10666 df-4 10667 df-5 10668 df-6 10669 df-7 10670 df-8 10671 df-9 10672 df-10 10673 df-n0 10867 df-z 10935 df-dec 11049 df-uz 11157 df-q 11262 df-rp 11300 df-xneg 11406 df-xadd 11407 df-xmul 11408 df-ioo 11636 df-icc 11639 df-fz 11782 df-fzo 11913 df-seq 12211 df-exp 12270 df-hash 12513 df-cj 13155 df-re 13156 df-im 13157 df-sqrt 13291 df-abs 13292 df-struct 15116 df-ndx 15117 df-slot 15118 df-base 15119 df-sets 15120 df-ress 15121 df-plusg 15196 df-mulr 15197 df-starv 15198 df-sca 15199 df-vsca 15200 df-ip 15201 df-tset 15202 df-ple 15203 df-ds 15205 df-unif 15206 df-hom 15207 df-cco 15208 df-rest 15314 df-topn 15315 df-0g 15333 df-gsum 15334 df-topgen 15335 df-pt 15336 df-prds 15339 df-xrs 15393 df-qtop 15399 df-imas 15400 df-qus 15402 df-xps 15403 df-mre 15485 df-mrc 15486 df-acs 15488 df-mgm 16481 df-sgrp 16520 df-mnd 16530 df-submnd 16576 df-grp 16666 df-mulg 16669 df-ghm 16874 df-gim 16916 df-gic 16917 df-cntz 16964 df-cmn 17425 df-psmet 18955 df-xmet 18956 df-met 18957 df-bl 18958 df-mopn 18959 df-cnfld 18964 df-top 19914 df-bases 19915 df-topon 19916 df-topsp 19917 df-cld 20027 df-cn 20236 df-cnp 20237 df-tx 20570 df-hmeo 20763 df-xms 21328 df-ms 21329 df-tms 21330 df-ii 21902 df-htpy 21994 df-phtpy 21995 df-phtpc 22016 df-pco 22029 df-om1 22030 df-pi1 22032 df-pcon 29937 |
This theorem is referenced by: sconpi1 29955 |
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