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| 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 999 |
. . . . . 6
#
# | |
| 2 | simp2 998 |
. . . . . . 7
# | |
| 3 | 2 | mul02d 8348 |
. . . . . 6
#  |
| 4 | 1, 3 | breqtrrd 4031 |
. . . . 5
#
# |
| 5 | simp1 997 |
. . . . . 6
# | |
| 6 | 0cnd 7949 |
. . . . . 6
# | |
| 7 | mulext 8570 |
. . . . . 6
#
#  # | |
| 8 | 5, 2, 6, 2, 7 | syl22anc 1239 |
. . . . 5
# # #
# |
| 9 | 4, 8 | mpd 13 |
. . . 4
# #
# |
| 10 | 9 | orcomd 729 |
. . 3
# #
# |
| 11 | apirr 8561 |
. . . 4
 # | |
| 12 | biorf 744 |
. . . 4
# #
 #
# | |
| 13 | 2, 11, 12 | 3syl 17 |
. . 3
# #
#
# |
| 14 | 10, 13 | mpbird 167 |
. 2
# # |
| 15 | 5 | mul01d 8349 |
. . . . 5
#
|
| 16 | 1, 15 | breqtrrd 4031 |
. . . 4
#
#
|
| 17 | mulext 8570 |
. . . . 5
#
#
# | |
| 18 | 5, 2, 5, 6, 17 | syl22anc 1239 |
. . . 4
# # #
# |
| 19 | 16, 18 | mpd 13 |
. . 3
# #
# |
| 20 | apirr 8561 |
. . . 4
# | |
| 21 | biorf 744 |
. . . 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 708 w3a 978 wcel 2148 class class class wbr 4003
(class class class)co 5874 cc 7808 cc0 7810 cmul 7815 # cap 8537 |
| 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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 |
| This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-pnf 7993 df-mnf 7994 df-ltxr 7996 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 |
| This theorem is referenced by: msqge0 8572 mulge0 8575 mulap0b 8611 |
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