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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 13328 | . . . . 5 β’ β = (Baseββfld) | |
2 | cnfldadd 13329 | . . . . 5 β’ + = (+gββfld) | |
3 | eqid 2177 | . . . . 5 β’ (invgββfld) = (invgββfld) | |
4 | eqid 2177 | . . . . 5 β’ (-gββfld) = (-gββfld) | |
5 | 1, 2, 3, 4 | grpsubval 12851 | . . . 4 β’ ((π₯ β β β§ π¦ β β) β (π₯(-gββfld)π¦) = (π₯ + ((invgββfld)βπ¦))) |
6 | cnfldneg 13336 | . . . . . 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 5888 | . . . 4 β’ ((π₯ β β β§ π¦ β β) β (π₯ + ((invgββfld)βπ¦)) = (π₯ + -π¦)) |
9 | negsub 8201 | . . . 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 5937 | . 2 β’ (π₯ β β, π¦ β β β¦ (π₯ β π¦)) = (π₯ β β, π¦ β β β¦ (π₯(-gββfld)π¦)) |
12 | subf 8155 | . . 3 β’ β :(β Γ β)βΆβ | |
13 | ffn 5364 | . . 3 β’ ( β :(β Γ β)βΆβ β β Fn (β Γ β)) | |
14 | fnovim 5980 | . . 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 13333 | . . . . 5 β’ βfld β Ring | |
17 | ringgrp 13115 | . . . . 5 β’ (βfld β Ring β βfld β Grp) | |
18 | 16, 17 | ax-mp 5 | . . . 4 β’ βfld β Grp |
19 | 1, 4 | grpsubf 12881 | . . . 4 β’ (βfld β Grp β (-gββfld):(β Γ β)βΆβ) |
20 | ffn 5364 | . . . 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 5980 | . . 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 4623 Fn wfn 5210 βΆwf 5211 βcfv 5215 (class class class)co 5872 β cmpo 5874 βcc 7806 + caddc 7811 β cmin 8124 -cneg 8125 Grpcgrp 12809 invgcminusg 12810 -gcsg 12811 Ringcrg 13110 βfldccnfld 13324 |
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 4117 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-mulrcl 7907 ax-addcom 7908 ax-mulcom 7909 ax-addass 7910 ax-mulass 7911 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-1rid 7915 ax-0id 7916 ax-rnegex 7917 ax-precex 7918 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-apti 7923 ax-pre-ltadd 7924 ax-pre-mulgt0 7925 ax-addf 7930 ax-mulf 7931 |
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 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-sub 8126 df-neg 8127 df-reap 8528 df-inn 8916 df-2 8974 df-3 8975 df-4 8976 df-5 8977 df-6 8978 df-7 8979 df-8 8980 df-9 8981 df-n0 9173 df-z 9250 df-dec 9381 df-uz 9525 df-fz 10005 df-cj 10844 df-struct 12456 df-ndx 12457 df-slot 12458 df-base 12460 df-sets 12461 df-plusg 12541 df-mulr 12542 df-starv 12543 df-0g 12695 df-mgm 12707 df-sgrp 12740 df-mnd 12750 df-grp 12812 df-minusg 12813 df-sbg 12814 df-cmn 13021 df-mgp 13062 df-ring 13112 df-cring 13113 df-icnfld 13325 |
This theorem is referenced by: zringsubgval 13364 |
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