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Mirrors > Home > ILE Home > Th. List > ofnegsub | Unicode version |
Description: Function analogue of negsub 8257. (Contributed by Mario Carneiro, 24-Jul-2014.) |
Ref | Expression |
---|---|
ofnegsub |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcl 7987 |
. . 3
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2 | 1 | adantl 277 |
. 2
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3 | simp2 1000 |
. 2
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4 | mulcl 7989 |
. . . 4
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5 | 4 | adantl 277 |
. . 3
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6 | ax-1cn 7955 |
. . . . . 6
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7 | 6 | negcli 8277 |
. . . . 5
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8 | 7 | fconst6 5445 |
. . . 4
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9 | 8 | a1i 9 |
. . 3
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10 | simp3 1001 |
. . 3
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11 | simp1 999 |
. . 3
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12 | inidm 3368 |
. . 3
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13 | 5, 9, 10, 11, 11, 12 | off 6135 |
. 2
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14 | subcl 8208 |
. . . 4
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15 | 14 | adantl 277 |
. . 3
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16 | 15, 3, 10, 11, 11, 12 | off 6135 |
. 2
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17 | eqidd 2194 |
. 2
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18 | 7 | a1i 9 |
. . . 4
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19 | 10 | ffnd 5396 |
. . . 4
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20 | eqidd 2194 |
. . . 4
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21 | 7 | a1i 9 |
. . . . 5
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22 | 10 | ffvelcdmda 5685 |
. . . . 5
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23 | 21, 22 | mulcld 8030 |
. . . 4
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24 | 11, 18, 19, 20, 23 | ofc1g 6143 |
. . 3
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25 | 22 | mulm1d 8419 |
. . 3
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26 | 24, 25 | eqtrd 2226 |
. 2
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27 | 3 | ffvelcdmda 5685 |
. . . 4
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28 | 27, 22 | negsubd 8326 |
. . 3
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29 | 3 | ffnd 5396 |
. . . 4
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30 | 27, 22 | subcld 8320 |
. . . 4
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31 | 29, 19, 11, 11, 12, 17, 20, 30 | ofvalg 6132 |
. . 3
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32 | 28, 31 | eqtr4d 2229 |
. 2
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33 | 2, 3, 13, 11, 11, 12, 16, 17, 26, 32 | offeq 6136 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-setind 4565 ax-resscn 7954 ax-1cn 7955 ax-icn 7957 ax-addcl 7958 ax-addrcl 7959 ax-mulcl 7960 ax-addcom 7962 ax-mulcom 7963 ax-addass 7964 ax-mulass 7965 ax-distr 7966 ax-i2m1 7967 ax-1rid 7969 ax-0id 7970 ax-rnegex 7971 ax-cnre 7973 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4322 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-iota 5207 df-fun 5248 df-fn 5249 df-f 5250 df-f1 5251 df-fo 5252 df-f1o 5253 df-fv 5254 df-riota 5865 df-ov 5913 df-oprab 5914 df-mpo 5915 df-of 6122 df-sub 8182 df-neg 8183 |
This theorem is referenced by: (None) |
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