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Mirrors > Home > ILE Home > Th. List > odzval | Unicode version |
Description: Value of the order function. This is a function of functions; the inner argument selects the base (i.e., mod for some , often prime) and the outer argument selects the integer or equivalence class (if you want to think about it that way) from the integers mod . In order to ensure the supremum is well-defined, we only define the expression when and are coprime. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by AV, 26-Sep-2020.) |
Ref | Expression |
---|---|
odzval | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5835 | . . . . . . . . 9 | |
2 | 1 | eqeq1d 2166 | . . . . . . . 8 |
3 | 2 | rabbidv 2701 | . . . . . . 7 |
4 | oveq1 5834 | . . . . . . . . 9 | |
5 | 4 | eqeq1d 2166 | . . . . . . . 8 |
6 | 5 | cbvrabv 2711 | . . . . . . 7 |
7 | 3, 6 | eqtr4di 2208 | . . . . . 6 |
8 | breq1 3970 | . . . . . . . 8 | |
9 | 8 | rabbidv 2701 | . . . . . . 7 |
10 | 9 | infeq1d 6959 | . . . . . 6 inf inf |
11 | 7, 10 | mpteq12dv 4049 | . . . . 5 inf inf |
12 | df-odz 12101 | . . . . 5 inf | |
13 | zex 9182 | . . . . . 6 | |
14 | 13 | mptrabex 5698 | . . . . 5 inf |
15 | 11, 12, 14 | fvmpt 5548 | . . . 4 inf |
16 | 15 | fveq1d 5473 | . . 3 inf |
17 | oveq1 5834 | . . . . . 6 | |
18 | 17 | eqeq1d 2166 | . . . . 5 |
19 | 18 | elrab 2868 | . . . 4 |
20 | oveq1 5834 | . . . . . . . . 9 | |
21 | 20 | oveq1d 5842 | . . . . . . . 8 |
22 | 21 | breq2d 3979 | . . . . . . 7 |
23 | 22 | rabbidv 2701 | . . . . . 6 |
24 | 23 | infeq1d 6959 | . . . . 5 inf inf |
25 | eqid 2157 | . . . . 5 inf inf | |
26 | reex 7869 | . . . . . 6 | |
27 | infex2g 6981 | . . . . . 6 inf | |
28 | 26, 27 | ax-mp 5 | . . . . 5 inf |
29 | 24, 25, 28 | fvmpt 5548 | . . . 4 inf inf |
30 | 19, 29 | sylbir 134 | . . 3 inf inf |
31 | 16, 30 | sylan9eq 2210 | . 2 inf |
32 | 31 | 3impb 1181 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 963 wceq 1335 wcel 2128 crab 2439 cvv 2712 class class class wbr 3967 cmpt 4028 cfv 5173 (class class class)co 5827 infcinf 6930 cr 7734 c1 7736 clt 7915 cmin 8051 cn 8839 cz 9173 cexp 10428 cdvds 11695 cgcd 11842 codz 12099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4082 ax-sep 4085 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-cnex 7826 ax-resscn 7827 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-iun 3853 df-br 3968 df-opab 4029 df-mpt 4030 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-f1 5178 df-fo 5179 df-f1o 5180 df-fv 5181 df-ov 5830 df-sup 6931 df-inf 6932 df-neg 8054 df-z 9174 df-odz 12101 |
This theorem is referenced by: odzcllem 12133 odzdvds 12136 |
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