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Mirrors > Home > ILE Home > Th. List > cnfldsub | GIF version |
Description: The subtraction operator in the field of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
cnfldsub | β’ β = (-gββfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfldbas 13350 | . . . . 5 β’ β = (Baseββfld) | |
2 | cnfldadd 13351 | . . . . 5 β’ + = (+gββfld) | |
3 | eqid 2177 | . . . . 5 β’ (invgββfld) = (invgββfld) | |
4 | eqid 2177 | . . . . 5 β’ (-gββfld) = (-gββfld) | |
5 | 1, 2, 3, 4 | grpsubval 12873 | . . . 4 β’ ((π₯ β β β§ π¦ β β) β (π₯(-gββfld)π¦) = (π₯ + ((invgββfld)βπ¦))) |
6 | cnfldneg 13358 | . . . . . 6 β’ (π¦ β β β ((invgββfld)βπ¦) = -π¦) | |
7 | 6 | adantl 277 | . . . . 5 β’ ((π₯ β β β§ π¦ β β) β ((invgββfld)βπ¦) = -π¦) |
8 | 7 | oveq2d 5890 | . . . 4 β’ ((π₯ β β β§ π¦ β β) β (π₯ + ((invgββfld)βπ¦)) = (π₯ + -π¦)) |
9 | negsub 8203 | . . . 4 β’ ((π₯ β β β§ π¦ β β) β (π₯ + -π¦) = (π₯ β π¦)) | |
10 | 5, 8, 9 | 3eqtrrd 2215 | . . 3 β’ ((π₯ β β β§ π¦ β β) β (π₯ β π¦) = (π₯(-gββfld)π¦)) |
11 | 10 | mpoeq3ia 5939 | . 2 β’ (π₯ β β, π¦ β β β¦ (π₯ β π¦)) = (π₯ β β, π¦ β β β¦ (π₯(-gββfld)π¦)) |
12 | subf 8157 | . . 3 β’ β :(β Γ β)βΆβ | |
13 | ffn 5365 | . . 3 β’ ( β :(β Γ β)βΆβ β β Fn (β Γ β)) | |
14 | fnovim 5982 | . . 3 β’ ( β Fn (β Γ β) β β = (π₯ β β, π¦ β β β¦ (π₯ β π¦))) | |
15 | 12, 13, 14 | mp2b 8 | . 2 β’ β = (π₯ β β, π¦ β β β¦ (π₯ β π¦)) |
16 | cnring 13355 | . . . . 5 β’ βfld β Ring | |
17 | ringgrp 13137 | . . . . 5 β’ (βfld β Ring β βfld β Grp) | |
18 | 16, 17 | ax-mp 5 | . . . 4 β’ βfld β Grp |
19 | 1, 4 | grpsubf 12903 | . . . 4 β’ (βfld β Grp β (-gββfld):(β Γ β)βΆβ) |
20 | ffn 5365 | . . . 4 β’ ((-gββfld):(β Γ β)βΆβ β (-gββfld) Fn (β Γ β)) | |
21 | 18, 19, 20 | mp2b 8 | . . 3 β’ (-gββfld) Fn (β Γ β) |
22 | fnovim 5982 | . . 3 β’ ((-gββfld) Fn (β Γ β) β (-gββfld) = (π₯ β β, π¦ β β β¦ (π₯(-gββfld)π¦))) | |
23 | 21, 22 | ax-mp 5 | . 2 β’ (-gββfld) = (π₯ β β, π¦ β β β¦ (π₯(-gββfld)π¦)) |
24 | 11, 15, 23 | 3eqtr4i 2208 | 1 β’ β = (-gββfld) |
Colors of variables: wff set class |
Syntax hints: β§ wa 104 = wceq 1353 β wcel 2148 Γ cxp 4624 Fn wfn 5211 βΆwf 5212 βcfv 5216 (class class class)co 5874 β cmpo 5876 βcc 7808 + caddc 7813 β cmin 8126 -cneg 8127 Grpcgrp 12831 invgcminusg 12832 -gcsg 12833 Ringcrg 13132 βfldccnfld 13346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-addf 7932 ax-mulf 7933 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-tp 3600 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-5 8979 df-6 8980 df-7 8981 df-8 8982 df-9 8983 df-n0 9175 df-z 9252 df-dec 9383 df-uz 9527 df-fz 10007 df-cj 10846 df-struct 12458 df-ndx 12459 df-slot 12460 df-base 12462 df-sets 12463 df-plusg 12543 df-mulr 12544 df-starv 12545 df-0g 12697 df-mgm 12729 df-sgrp 12762 df-mnd 12772 df-grp 12834 df-minusg 12835 df-sbg 12836 df-cmn 13043 df-mgp 13084 df-ring 13134 df-cring 13135 df-icnfld 13347 |
This theorem is referenced by: zringsubgval 13386 |
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