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| Mirrors > Home > ILE Home > Th. List > un0mulcl | Unicode version | ||
| Description: If  | 
| Ref | Expression | 
|---|---|
| un0addcl.1 | 
 | 
| un0addcl.2 | 
 | 
| un0mulcl.3 | 
 | 
| Ref | Expression | 
|---|---|
| un0mulcl | 
 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | un0addcl.2 | 
. . . . 5
 | |
| 2 | 1 | eleq2i 2263 | 
. . . 4
 | 
| 3 | elun 3304 | 
. . . 4
 | |
| 4 | 2, 3 | bitri 184 | 
. . 3
 | 
| 5 | 1 | eleq2i 2263 | 
. . . . . 6
 | 
| 6 | elun 3304 | 
. . . . . 6
 | |
| 7 | 5, 6 | bitri 184 | 
. . . . 5
 | 
| 8 | ssun1 3326 | 
. . . . . . . . 9
 | |
| 9 | 8, 1 | sseqtrri 3218 | 
. . . . . . . 8
 | 
| 10 | un0mulcl.3 | 
. . . . . . . 8
 | |
| 11 | 9, 10 | sselid 3181 | 
. . . . . . 7
 | 
| 12 | 11 | expr 375 | 
. . . . . 6
 | 
| 13 | un0addcl.1 | 
. . . . . . . . . . 11
 | |
| 14 | 13 | sselda 3183 | 
. . . . . . . . . 10
 | 
| 15 | 14 | mul02d 8418 | 
. . . . . . . . 9
 | 
| 16 | ssun2 3327 | 
. . . . . . . . . . 11
 | |
| 17 | 16, 1 | sseqtrri 3218 | 
. . . . . . . . . 10
 | 
| 18 | c0ex 8020 | 
. . . . . . . . . . 11
 | |
| 19 | 18 | snss 3757 | 
. . . . . . . . . 10
 | 
| 20 | 17, 19 | mpbir 146 | 
. . . . . . . . 9
 | 
| 21 | 15, 20 | eqeltrdi 2287 | 
. . . . . . . 8
 | 
| 22 | elsni 3640 | 
. . . . . . . . . 10
 | |
| 23 | 22 | oveq1d 5937 | 
. . . . . . . . 9
 | 
| 24 | 23 | eleq1d 2265 | 
. . . . . . . 8
 | 
| 25 | 21, 24 | syl5ibrcom 157 | 
. . . . . . 7
 | 
| 26 | 25 | impancom 260 | 
. . . . . 6
 | 
| 27 | 12, 26 | jaodan 798 | 
. . . . 5
 | 
| 28 | 7, 27 | sylan2b 287 | 
. . . 4
 | 
| 29 | 0cnd 8019 | 
. . . . . . . . . . 11
 | |
| 30 | 29 | snssd 3767 | 
. . . . . . . . . 10
 | 
| 31 | 13, 30 | unssd 3339 | 
. . . . . . . . 9
 | 
| 32 | 1, 31 | eqsstrid 3229 | 
. . . . . . . 8
 | 
| 33 | 32 | sselda 3183 | 
. . . . . . 7
 | 
| 34 | 33 | mul01d 8419 | 
. . . . . 6
 | 
| 35 | 34, 20 | eqeltrdi 2287 | 
. . . . 5
 | 
| 36 | elsni 3640 | 
. . . . . . 7
 | |
| 37 | 36 | oveq2d 5938 | 
. . . . . 6
 | 
| 38 | 37 | eleq1d 2265 | 
. . . . 5
 | 
| 39 | 35, 38 | syl5ibrcom 157 | 
. . . 4
 | 
| 40 | 28, 39 | jaod 718 | 
. . 3
 | 
| 41 | 4, 40 | biimtrid 152 | 
. 2
 | 
| 42 | 41 | impr 379 | 
1
 | 
| 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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-setind 4573 ax-resscn 7971 ax-1cn 7972 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-distr 7983 ax-i2m1 7984 ax-0id 7987 ax-rnegex 7988 ax-cnre 7990 | 
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-br 4034 df-opab 4095 df-id 4328 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-iota 5219 df-fun 5260 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-sub 8199 | 
| This theorem is referenced by: nn0mulcl 9285 plymullem 14986 | 
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