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Mirrors > Home > ILE Home > Th. List > qusval | Unicode version |
Description: Value of a quotient structure. (Contributed by Mario Carneiro, 23-Feb-2015.) |
Ref | Expression |
---|---|
qusval.u |
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qusval.v |
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qusval.f |
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qusval.e |
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qusval.r |
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Ref | Expression |
---|---|
qusval |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qusval.u |
. 2
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2 | df-qus 12746 |
. . . 4
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3 | 2 | a1i 9 |
. . 3
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4 | simprl 529 |
. . . . . . . 8
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5 | 4 | fveq2d 5534 |
. . . . . . 7
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6 | qusval.v |
. . . . . . . 8
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7 | 6 | adantr 276 |
. . . . . . 7
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8 | 5, 7 | eqtr4d 2225 |
. . . . . 6
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9 | eceq2 6590 |
. . . . . . 7
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10 | 9 | ad2antll 491 |
. . . . . 6
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11 | 8, 10 | mpteq12dv 4100 |
. . . . 5
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12 | qusval.f |
. . . . 5
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13 | 11, 12 | eqtr4di 2240 |
. . . 4
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14 | 13, 4 | oveq12d 5909 |
. . 3
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15 | qusval.r |
. . . 4
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16 | 15 | elexd 2765 |
. . 3
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17 | qusval.e |
. . . 4
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18 | 17 | elexd 2765 |
. . 3
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19 | basfn 12538 |
. . . . . . . 8
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20 | funfvex 5547 |
. . . . . . . . 9
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21 | 20 | funfni 5331 |
. . . . . . . 8
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22 | 19, 16, 21 | sylancr 414 |
. . . . . . 7
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23 | 6, 22 | eqeltrd 2266 |
. . . . . 6
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24 | 23 | mptexd 5759 |
. . . . 5
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25 | 12, 24 | eqeltrid 2276 |
. . . 4
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26 | imasex 12748 |
. . . 4
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27 | 25, 15, 26 | syl2anc 411 |
. . 3
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28 | 3, 14, 16, 18, 27 | ovmpod 6019 |
. 2
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29 | 1, 28 | eqtrd 2222 |
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-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7920 ax-resscn 7921 ax-1re 7923 ax-addrcl 7926 |
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 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-tp 3615 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-ec 6555 df-inn 8938 df-2 8996 df-3 8997 df-ndx 12483 df-slot 12484 df-base 12486 df-plusg 12568 df-mulr 12569 df-iimas 12745 df-qus 12746 |
This theorem is referenced by: qusin 12769 qusbas 12770 qusaddval 12777 qusaddf 12778 qusmulval 12779 qusmulf 12780 qusgrp2 13021 qusrng 13273 qusring2 13377 |
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