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Mirrors > Home > MPE Home > Th. List > mapdom3 | Structured version Visualization version Unicode version |
Description: Set exponentiation dominates the mantissa. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
mapdom3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 3931 | . . 3 | |
2 | oveq1 6657 | . . . . . . . . . 10 | |
3 | id 22 | . . . . . . . . . 10 | |
4 | 2, 3 | breq12d 4666 | . . . . . . . . 9 |
5 | vex 3203 | . . . . . . . . . 10 | |
6 | vex 3203 | . . . . . . . . . 10 | |
7 | 5, 6 | mapsnen 8035 | . . . . . . . . 9 |
8 | 4, 7 | vtoclg 3266 | . . . . . . . 8 |
9 | 8 | 3ad2ant1 1082 | . . . . . . 7 |
10 | 9 | ensymd 8007 | . . . . . 6 |
11 | simp2 1062 | . . . . . . . 8 | |
12 | simp3 1063 | . . . . . . . . 9 | |
13 | 12 | snssd 4340 | . . . . . . . 8 |
14 | ssdomg 8001 | . . . . . . . 8 | |
15 | 11, 13, 14 | sylc 65 | . . . . . . 7 |
16 | 6 | snnz 4309 | . . . . . . . 8 |
17 | simpl 473 | . . . . . . . . 9 | |
18 | 17 | necon3ai 2819 | . . . . . . . 8 |
19 | 16, 18 | ax-mp 5 | . . . . . . 7 |
20 | mapdom2 8131 | . . . . . . 7 | |
21 | 15, 19, 20 | sylancl 694 | . . . . . 6 |
22 | endomtr 8014 | . . . . . 6 | |
23 | 10, 21, 22 | syl2anc 693 | . . . . 5 |
24 | 23 | 3expia 1267 | . . . 4 |
25 | 24 | exlimdv 1861 | . . 3 |
26 | 1, 25 | syl5bi 232 | . 2 |
27 | 26 | 3impia 1261 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wa 384 w3a 1037 wceq 1483 wex 1704 wcel 1990 wne 2794 wss 3574 c0 3915 csn 4177 class class class wbr 4653 (class class class)co 6650 cmap 7857 cen 7952 cdom 7953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 |
This theorem is referenced by: infmap2 9040 |
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