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 1001 |
. . . . . 6
#
# | |
| 2 | simp2 1000 |
. . . . . . 7
# | |
| 3 | 2 | mul02d 8411 |
. . . . . 6
#  |
| 4 | 1, 3 | breqtrrd 4057 |
. . . . 5
#
# |
| 5 | simp1 999 |
. . . . . 6
# | |
| 6 | 0cnd 8012 |
. . . . . 6
# | |
| 7 | mulext 8633 |
. . . . . 6
#
#  # | |
| 8 | 5, 2, 6, 2, 7 | syl22anc 1250 |
. . . . 5
# # #
# |
| 9 | 4, 8 | mpd 13 |
. . . 4
# #
# |
| 10 | 9 | orcomd 730 |
. . 3
# #
# |
| 11 | apirr 8624 |
. . . 4
 # | |
| 12 | biorf 745 |
. . . 4
# #
 #
# | |
| 13 | 2, 11, 12 | 3syl 17 |
. . 3
# #
#
# |
| 14 | 10, 13 | mpbird 167 |
. 2
# # |
| 15 | 5 | mul01d 8412 |
. . . . 5
#
|
| 16 | 1, 15 | breqtrrd 4057 |
. . . 4
#
#
|
| 17 | mulext 8633 |
. . . . 5
#
#
# | |
| 18 | 5, 2, 5, 6, 17 | syl22anc 1250 |
. . . 4
# # #
# |
| 19 | 16, 18 | mpd 13 |
. . 3
# #
# |
| 20 | apirr 8624 |
. . . 4
# | |
| 21 | biorf 745 |
. . . 4
# #
#
# | |
| 22 | 5, 20, 21 | 3syl 17 |
. . 3
# #
#
# |
| 23 | 19, 22 | mpbird 167 |
. 2
# # |
| 24 | 14, 23 | jca 306 |
1
# #
# |
| Colors of variables: wff set class |
| Syntax hints: wn 3
wi 4
wa 104
wb 105 wo 709 w3a 980 wcel 2164 class class class wbr 4029
(class class class)co 5918 cc 7870 cc0 7872 cmul 7877 # cap 8600 |
| 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 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 |
| 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-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2986 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-br 4030 df-opab 4091 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-iota 5215 df-fun 5256 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-pnf 8056 df-mnf 8057 df-ltxr 8059 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 |
| This theorem is referenced by: msqge0 8635 mulge0 8638 mulap0b 8674 |
| Copyright terms: Public domain | W3C validator |