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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 983 |
. . . . . 6
#
# | |
| 2 | simp2 982 |
. . . . . . 7
# | |
| 3 | 2 | mul02d 8154 |
. . . . . 6
#  |
| 4 | 1, 3 | breqtrrd 3956 |
. . . . 5
#
# |
| 5 | simp1 981 |
. . . . . 6
# | |
| 6 | 0cnd 7759 |
. . . . . 6
# | |
| 7 | mulext 8376 |
. . . . . 6
#
#  # | |
| 8 | 5, 2, 6, 2, 7 | syl22anc 1217 |
. . . . 5
# # #
# |
| 9 | 4, 8 | mpd 13 |
. . . 4
# #
# |
| 10 | 9 | orcomd 718 |
. . 3
# #
# |
| 11 | apirr 8367 |
. . . 4
 # | |
| 12 | biorf 733 |
. . . 4
# #
 #
# | |
| 13 | 2, 11, 12 | 3syl 17 |
. . 3
# #
#
# |
| 14 | 10, 13 | mpbird 166 |
. 2
# # |
| 15 | 5 | mul01d 8155 |
. . . . 5
#
|
| 16 | 1, 15 | breqtrrd 3956 |
. . . 4
#
#
|
| 17 | mulext 8376 |
. . . . 5
#
#
# | |
| 18 | 5, 2, 5, 6, 17 | syl22anc 1217 |
. . . 4
# # #
# |
| 19 | 16, 18 | mpd 13 |
. . 3
# #
# |
| 20 | apirr 8367 |
. . . 4
# | |
| 21 | biorf 733 |
. . . 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 697 w3a 962 wcel 1480 class class class wbr 3929
(class class class)co 5774 cc 7618 cc0 7620 cmul 7625 # cap 8343 |
| 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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
| This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 |
| This theorem is referenced by: msqge0 8378 mulge0 8381 mulap0b 8416 |
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