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Mirrors > Home > MPE Home > Th. List > elpi1 | Structured version Unicode version |
Description: The elements of the fundamental group. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.) |
Ref | Expression |
---|---|
elpi1.g |
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elpi1.b |
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elpi1.1 |
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elpi1.2 |
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Ref | Expression |
---|---|
elpi1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpi1.g |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | elpi1.1 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | elpi1.2 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | elpi1.b |
. . . . 5
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5 | 4 | a1i 11 |
. . . 4
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6 | 1, 2, 3, 5 | pi1bas2 20746 |
. . 3
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7 | 6 | eleq2d 2524 |
. 2
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8 | elex 3087 |
. . . 4
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9 | id 22 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
10 | fvex 5810 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | ecexg 7216 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
12 | 10, 11 | ax-mp 5 |
. . . . . 6
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13 | 9, 12 | syl6eqel 2550 |
. . . . 5
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14 | 13 | rexlimivw 2943 |
. . . 4
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15 | elqsg 7263 |
. . . 4
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16 | 8, 14, 15 | pm5.21nii 353 |
. . 3
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17 | 1, 2, 3, 5 | pi1eluni 20747 |
. . . . . . 7
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18 | 3anass 969 |
. . . . . . 7
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19 | 17, 18 | syl6bb 261 |
. . . . . 6
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20 | 19 | anbi1d 704 |
. . . . 5
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21 | anass 649 |
. . . . 5
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22 | 20, 21 | syl6bb 261 |
. . . 4
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23 | 22 | rexbidv2 2858 |
. . 3
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24 | 16, 23 | syl5bb 257 |
. 2
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25 | 7, 24 | bitrd 253 |
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 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-rep 4512 ax-sep 4522 ax-nul 4530 ax-pow 4579 ax-pr 4640 ax-un 6483 ax-inf2 7959 ax-cnex 9450 ax-resscn 9451 ax-1cn 9452 ax-icn 9453 ax-addcl 9454 ax-addrcl 9455 ax-mulcl 9456 ax-mulrcl 9457 ax-mulcom 9458 ax-addass 9459 ax-mulass 9460 ax-distr 9461 ax-i2m1 9462 ax-1ne0 9463 ax-1rid 9464 ax-rnegex 9465 ax-rrecex 9466 ax-cnre 9467 ax-pre-lttri 9468 ax-pre-lttrn 9469 ax-pre-ltadd 9470 ax-pre-mulgt0 9471 ax-pre-sup 9472 ax-mulf 9474 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-reu 2806 df-rmo 2807 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3397 df-dif 3440 df-un 3442 df-in 3444 df-ss 3451 df-pss 3453 df-nul 3747 df-if 3901 df-pw 3971 df-sn 3987 df-pr 3989 df-tp 3991 df-op 3993 df-uni 4201 df-int 4238 df-iun 4282 df-iin 4283 df-br 4402 df-opab 4460 df-mpt 4461 df-tr 4495 df-eprel 4741 df-id 4745 df-po 4750 df-so 4751 df-fr 4788 df-se 4789 df-we 4790 df-ord 4831 df-on 4832 df-lim 4833 df-suc 4834 df-xp 4955 df-rel 4956 df-cnv 4957 df-co 4958 df-dm 4959 df-rn 4960 df-res 4961 df-ima 4962 df-iota 5490 df-fun 5529 df-fn 5530 df-f 5531 df-f1 5532 df-fo 5533 df-f1o 5534 df-fv 5535 df-isom 5536 df-riota 6162 df-ov 6204 df-oprab 6205 df-mpt2 6206 df-of 6431 df-om 6588 df-1st 6688 df-2nd 6689 df-supp 6802 df-recs 6943 df-rdg 6977 df-1o 7031 df-2o 7032 df-oadd 7035 df-er 7212 df-ec 7214 df-qs 7218 df-map 7327 df-ixp 7375 df-en 7422 df-dom 7423 df-sdom 7424 df-fin 7425 df-fsupp 7733 df-fi 7773 df-sup 7803 df-oi 7836 df-card 8221 df-cda 8449 df-pnf 9532 df-mnf 9533 df-xr 9534 df-ltxr 9535 df-le 9536 df-sub 9709 df-neg 9710 df-div 10106 df-nn 10435 df-2 10492 df-3 10493 df-4 10494 df-5 10495 df-6 10496 df-7 10497 df-8 10498 df-9 10499 df-10 10500 df-n0 10692 df-z 10759 df-dec 10868 df-uz 10974 df-q 11066 df-rp 11104 df-xneg 11201 df-xadd 11202 df-xmul 11203 df-ioo 11416 df-icc 11419 df-fz 11556 df-fzo 11667 df-seq 11925 df-exp 11984 df-hash 12222 df-cj 12707 df-re 12708 df-im 12709 df-sqr 12843 df-abs 12844 df-struct 14295 df-ndx 14296 df-slot 14297 df-base 14298 df-sets 14299 df-ress 14300 df-plusg 14371 df-mulr 14372 df-starv 14373 df-sca 14374 df-vsca 14375 df-ip 14376 df-tset 14377 df-ple 14378 df-ds 14380 df-unif 14381 df-hom 14382 df-cco 14383 df-rest 14481 df-topn 14482 df-0g 14500 df-gsum 14501 df-topgen 14502 df-pt 14503 df-prds 14506 df-xrs 14560 df-qtop 14565 df-imas 14566 df-divs 14567 df-xps 14568 df-mre 14644 df-mrc 14645 df-acs 14647 df-mnd 15535 df-submnd 15585 df-mulg 15668 df-cntz 15955 df-cmn 16401 df-psmet 17935 df-xmet 17936 df-met 17937 df-bl 17938 df-mopn 17939 df-cnfld 17945 df-top 18636 df-bases 18638 df-topon 18639 df-topsp 18640 df-cld 18756 df-cn 18964 df-cnp 18965 df-tx 19268 df-hmeo 19461 df-xms 20028 df-ms 20029 df-tms 20030 df-ii 20586 df-htpy 20675 df-phtpy 20676 df-phtpc 20697 df-om1 20711 df-pi1 20713 |
This theorem is referenced by: elpi1i 20751 sconpi1 27273 |
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