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Mirrors > Home > HSE Home > Th. List > dmdi4 | Structured version Visualization version Unicode version |
Description: Consequence of the dual modular pair property. (Contributed by NM, 14-Jan-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dmdi4 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmdbr4 27971 |
. . . 4
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2 | 1 | biimpd 212 |
. . 3
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3 | oveq1 6283 |
. . . . . 6
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4 | 3 | ineq1d 3601 |
. . . . 5
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5 | 3 | ineq1d 3601 |
. . . . . 6
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6 | 5 | oveq1d 6291 |
. . . . 5
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7 | 4, 6 | sseq12d 3429 |
. . . 4
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8 | 7 | rspcv 3114 |
. . 3
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9 | 2, 8 | sylan9 667 |
. 2
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10 | 9 | 3impa 1205 |
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 1673 ax-4 1686 ax-5 1762 ax-6 1809 ax-7 1855 ax-8 1893 ax-9 1900 ax-10 1919 ax-11 1924 ax-12 1937 ax-13 2092 ax-ext 2432 ax-rep 4487 ax-sep 4497 ax-nul 4506 ax-pow 4554 ax-pr 4612 ax-un 6571 ax-inf2 8133 ax-cc 8852 ax-cnex 9582 ax-resscn 9583 ax-1cn 9584 ax-icn 9585 ax-addcl 9586 ax-addrcl 9587 ax-mulcl 9588 ax-mulrcl 9589 ax-mulcom 9590 ax-addass 9591 ax-mulass 9592 ax-distr 9593 ax-i2m1 9594 ax-1ne0 9595 ax-1rid 9596 ax-rnegex 9597 ax-rrecex 9598 ax-cnre 9599 ax-pre-lttri 9600 ax-pre-lttrn 9601 ax-pre-ltadd 9602 ax-pre-mulgt0 9603 ax-pre-sup 9604 ax-addf 9605 ax-mulf 9606 ax-hilex 26664 ax-hfvadd 26665 ax-hvcom 26666 ax-hvass 26667 ax-hv0cl 26668 ax-hvaddid 26669 ax-hfvmul 26670 ax-hvmulid 26671 ax-hvmulass 26672 ax-hvdistr1 26673 ax-hvdistr2 26674 ax-hvmul0 26675 ax-hfi 26744 ax-his1 26747 ax-his2 26748 ax-his3 26749 ax-his4 26750 ax-hcompl 26867 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 987 df-3an 988 df-tru 1451 df-fal 1454 df-ex 1668 df-nf 1672 df-sb 1802 df-eu 2304 df-mo 2305 df-clab 2439 df-cleq 2445 df-clel 2448 df-nfc 2582 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rmo 2745 df-rab 2746 df-v 3015 df-sbc 3236 df-csb 3332 df-dif 3375 df-un 3377 df-in 3379 df-ss 3386 df-pss 3388 df-nul 3700 df-if 3850 df-pw 3921 df-sn 3937 df-pr 3939 df-tp 3941 df-op 3943 df-uni 4169 df-int 4205 df-iun 4250 df-iin 4251 df-br 4375 df-opab 4434 df-mpt 4435 df-tr 4470 df-eprel 4723 df-id 4727 df-po 4733 df-so 4734 df-fr 4771 df-se 4772 df-we 4773 df-xp 4818 df-rel 4819 df-cnv 4820 df-co 4821 df-dm 4822 df-rn 4823 df-res 4824 df-ima 4825 df-pred 5359 df-ord 5405 df-on 5406 df-lim 5407 df-suc 5408 df-iota 5525 df-fun 5563 df-fn 5564 df-f 5565 df-f1 5566 df-fo 5567 df-f1o 5568 df-fv 5569 df-isom 5570 df-riota 6238 df-ov 6279 df-oprab 6280 df-mpt2 6281 df-of 6519 df-om 6681 df-1st 6781 df-2nd 6782 df-supp 6903 df-wrecs 7015 df-recs 7077 df-rdg 7115 df-1o 7169 df-2o 7170 df-oadd 7173 df-omul 7174 df-er 7350 df-map 7461 df-pm 7462 df-ixp 7510 df-en 7557 df-dom 7558 df-sdom 7559 df-fin 7560 df-fsupp 7871 df-fi 7912 df-sup 7943 df-inf 7944 df-oi 8012 df-card 8360 df-acn 8363 df-cda 8585 df-pnf 9664 df-mnf 9665 df-xr 9666 df-ltxr 9667 df-le 9668 df-sub 9849 df-neg 9850 df-div 10259 df-nn 10599 df-2 10657 df-3 10658 df-4 10659 df-5 10660 df-6 10661 df-7 10662 df-8 10663 df-9 10664 df-10 10665 df-n0 10860 df-z 10928 df-dec 11042 df-uz 11150 df-q 11255 df-rp 11293 df-xneg 11399 df-xadd 11400 df-xmul 11401 df-ioo 11629 df-ico 11631 df-icc 11632 df-fz 11776 df-fzo 11909 df-fl 12022 df-seq 12208 df-exp 12267 df-hash 12510 df-cj 13173 df-re 13174 df-im 13175 df-sqrt 13309 df-abs 13310 df-clim 13563 df-rlim 13564 df-sum 13764 df-struct 15134 df-ndx 15135 df-slot 15136 df-base 15137 df-sets 15138 df-ress 15139 df-plusg 15214 df-mulr 15215 df-starv 15216 df-sca 15217 df-vsca 15218 df-ip 15219 df-tset 15220 df-ple 15221 df-ds 15223 df-unif 15224 df-hom 15225 df-cco 15226 df-rest 15332 df-topn 15333 df-0g 15351 df-gsum 15352 df-topgen 15353 df-pt 15354 df-prds 15357 df-xrs 15411 df-qtop 15417 df-imas 15418 df-xps 15421 df-mre 15503 df-mrc 15504 df-acs 15506 df-mgm 16499 df-sgrp 16538 df-mnd 16548 df-submnd 16594 df-mulg 16687 df-cntz 16982 df-cmn 17443 df-psmet 18973 df-xmet 18974 df-met 18975 df-bl 18976 df-mopn 18977 df-fbas 18978 df-fg 18979 df-cnfld 18982 df-top 19932 df-bases 19933 df-topon 19934 df-topsp 19935 df-cld 20045 df-ntr 20046 df-cls 20047 df-nei 20125 df-cn 20254 df-cnp 20255 df-lm 20256 df-haus 20342 df-tx 20588 df-hmeo 20781 df-fil 20872 df-fm 20964 df-flim 20965 df-flf 20966 df-xms 21346 df-ms 21347 df-tms 21348 df-cfil 22236 df-cau 22237 df-cmet 22238 df-grpo 25931 df-gid 25932 df-ginv 25933 df-gdiv 25934 df-ablo 26022 df-subgo 26042 df-vc 26177 df-nv 26223 df-va 26226 df-ba 26227 df-sm 26228 df-0v 26229 df-vs 26230 df-nmcv 26231 df-ims 26232 df-dip 26349 df-ssp 26373 df-ph 26466 df-cbn 26517 df-hnorm 26633 df-hba 26634 df-hvsub 26636 df-hlim 26637 df-hcau 26638 df-sh 26872 df-ch 26886 df-oc 26917 df-ch0 26918 df-shs 26973 df-chj 26975 df-dmd 27946 |
This theorem is referenced by: dmdbr5ati 28087 |
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