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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > qqhf | Structured version Visualization version Unicode version |
Description: QQHom as a function. (Contributed by Thierry Arnoux, 28-Oct-2017.) |
Ref | Expression |
---|---|
qqhval2.0 |
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qqhval2.1 |
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qqhval2.2 |
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Ref | Expression |
---|---|
qqhf |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qqhval2.0 |
. . 3
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2 | qqhval2.1 |
. . 3
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3 | qqhval2.2 |
. . 3
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4 | 1, 2, 3 | qqhval2 28779 |
. 2
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5 | drngring 17975 |
. . . . 5
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6 | 5 | adantr 467 |
. . . 4
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7 | 6 | adantr 467 |
. . 3
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8 | 3 | zrhrhm 19076 |
. . . . 5
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9 | zringbas 19038 |
. . . . . 6
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10 | 9, 1 | rhmf 17947 |
. . . . 5
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11 | 7, 8, 10 | 3syl 18 |
. . . 4
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12 | qnumcl 14682 |
. . . . 5
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13 | 12 | adantl 468 |
. . . 4
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14 | 11, 13 | ffvelrnd 6021 |
. . 3
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15 | simpll 759 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | qdencl 14683 |
. . . . . . 7
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17 | 16 | adantl 468 |
. . . . . 6
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18 | 17 | nnzd 11036 |
. . . . 5
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19 | 11, 18 | ffvelrnd 6021 |
. . . 4
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20 | 17 | nnne0d 10651 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 20 | neneqd 2628 |
. . . . . . . . 9
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22 | fvex 5873 |
. . . . . . . . . 10
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23 | 22 | elsnc 3991 |
. . . . . . . . 9
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24 | 21, 23 | sylnibr 307 |
. . . . . . . 8
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25 | eqid 2450 |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
26 | 1, 3, 25 | zrhker 28774 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 26 | biimpa 487 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 5, 27 | sylan 474 |
. . . . . . . . 9
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29 | 28 | adantr 467 |
. . . . . . . 8
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30 | 24, 29 | neleqtrrd 2550 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | ffn 5726 |
. . . . . . . . . . . 12
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32 | 8, 10, 31 | 3syl 18 |
. . . . . . . . . . 11
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33 | elpreima 6000 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
34 | 5, 32, 33 | 3syl 18 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
35 | 34 | biimpar 488 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
36 | 35 | expr 619 |
. . . . . . . 8
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37 | 36 | con3dimp 443 |
. . . . . . 7
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38 | 15, 18, 30, 37 | syl21anc 1266 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
39 | fvex 5873 |
. . . . . . 7
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40 | 39 | elsnc 3991 |
. . . . . 6
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41 | 38, 40 | sylnib 306 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
42 | 41 | neqned 2630 |
. . . 4
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43 | eqid 2450 |
. . . . . 6
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44 | 1, 43, 25 | drngunit 17973 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
45 | 44 | biimpar 488 |
. . . 4
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46 | 15, 19, 42, 45 | syl12anc 1265 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
47 | 1, 43, 2 | dvrcl 17907 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
48 | 7, 14, 46, 47 | syl3anc 1267 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
49 | 4, 48 | fmpt3d 6045 |
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-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-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-riota 6250 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-om 6690 df-1st 6790 df-2nd 6791 df-tpos 6970 df-wrecs 7025 df-recs 7087 df-rdg 7125 df-1o 7179 df-oadd 7183 df-er 7360 df-map 7471 df-en 7567 df-dom 7568 df-sdom 7569 df-fin 7570 df-sup 7953 df-inf 7954 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-fz 11782 df-fl 12025 df-mod 12094 df-seq 12211 df-exp 12270 df-cj 13155 df-re 13156 df-im 13157 df-sqrt 13291 df-abs 13292 df-dvds 14299 df-gcd 14462 df-numer 14677 df-denom 14678 df-gz 14867 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-tset 15202 df-ple 15203 df-ds 15205 df-unif 15206 df-0g 15333 df-mgm 16481 df-sgrp 16520 df-mnd 16530 df-mhm 16575 df-grp 16666 df-minusg 16667 df-sbg 16668 df-mulg 16669 df-subg 16807 df-ghm 16874 df-od 17165 df-cmn 17425 df-mgp 17717 df-ur 17729 df-ring 17775 df-cring 17776 df-oppr 17844 df-dvdsr 17862 df-unit 17863 df-invr 17893 df-dvr 17904 df-rnghom 17936 df-drng 17970 df-subrg 17999 df-cnfld 18964 df-zring 19033 df-zrh 19068 df-chr 19070 df-qqh 28770 |
This theorem is referenced by: qqhghm 28785 qqhrhm 28786 qqhcn 28788 qqhucn 28789 qqhre 28817 |
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