Nearby theorems
|
|
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > ILE Home > Th. List > mulap0r | Unicode version | ||
| Description: A product apart from zero. Lemma 2.13 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 24-Feb-2020.) |
| Ref | Expression |
|---|---|
| mulap0r |
     ![/\](wedge.gif "/\"){kind=small} 
 #   #
# |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp3 989 |
. . . . . 6
#
# | |
| 2 | simp2 988 |
. . . . . . 7
# | |
| 3 | 2 | mul02d 8290 |
. . . . . 6
#  |
| 4 | 1, 3 | breqtrrd 4010 |
. . . . 5
#
# |
| 5 | simp1 987 |
. . . . . 6
# | |
| 6 | 0cnd 7892 |
. . . . . 6
# | |
| 7 | mulext 8512 |
. . . . . 6
#
#  # | |
| 8 | 5, 2, 6, 2, 7 | syl22anc 1229 |
. . . . 5
# # #
# |
| 9 | 4, 8 | mpd 13 |
. . . 4
# #
# |
| 10 | 9 | orcomd 719 |
. . 3
# #
# |
| 11 | apirr 8503 |
. . . 4
 # | |
| 12 | biorf 734 |
. . . 4
# #
 #
# | |
| 13 | 2, 11, 12 | 3syl 17 |
. . 3
# #
#
# |
| 14 | 10, 13 | mpbird 166 |
. 2
# # |
| 15 | 5 | mul01d 8291 |
. . . . 5
#
|
| 16 | 1, 15 | breqtrrd 4010 |
. . . 4
#
#
|
| 17 | mulext 8512 |
. . . . 5
#
#
# | |
| 18 | 5, 2, 5, 6, 17 | syl22anc 1229 |
. . . 4
# # #
# |
| 19 | 16, 18 | mpd 13 |
. . 3
# #
# |
| 20 | apirr 8503 |
. . . 4
# | |
| 21 | biorf 734 |
. . . 4
# #
#
# | |
| 22 | 5, 20, 21 | 3syl 17 |
. . 3
# #
#
# |
| 23 | 19, 22 | mpbird 166 |
. 2
# # |
| 24 | 14, 23 | jca 304 |
1
# #
# |
| Colors of variables: wff set class |
| Syntax hints: wn 3
wi 4
wa 103
wb 104 wo 698 w3a 968 wcel 2136 class class class wbr 3982
(class class class)co 5842 cc 7751 cc0 7753 cmul 7758 # cap 8479 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 |
| This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pnf 7935 df-mnf 7936 df-ltxr 7938 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 |
| This theorem is referenced by: msqge0 8514 mulge0 8517 mulap0b 8552 |
| Copyright terms: Public domain | W3C validator |