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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 984 |
. . . . . 6
#
# | |
| 2 | simp2 983 |
. . . . . . 7
# | |
| 3 | 2 | mul02d 8246 |
. . . . . 6
#  |
| 4 | 1, 3 | breqtrrd 3988 |
. . . . 5
#
# |
| 5 | simp1 982 |
. . . . . 6
# | |
| 6 | 0cnd 7850 |
. . . . . 6
# | |
| 7 | mulext 8468 |
. . . . . 6
#
#  # | |
| 8 | 5, 2, 6, 2, 7 | syl22anc 1218 |
. . . . 5
# # #
# |
| 9 | 4, 8 | mpd 13 |
. . . 4
# #
# |
| 10 | 9 | orcomd 719 |
. . 3
# #
# |
| 11 | apirr 8459 |
. . . 4
 # | |
| 12 | biorf 734 |
. . . 4
# #
 #
# | |
| 13 | 2, 11, 12 | 3syl 17 |
. . 3
# #
#
# |
| 14 | 10, 13 | mpbird 166 |
. 2
# # |
| 15 | 5 | mul01d 8247 |
. . . . 5
#
|
| 16 | 1, 15 | breqtrrd 3988 |
. . . 4
#
#
|
| 17 | mulext 8468 |
. . . . 5
#
#
# | |
| 18 | 5, 2, 5, 6, 17 | syl22anc 1218 |
. . . 4
# # #
# |
| 19 | 16, 18 | mpd 13 |
. . 3
# #
# |
| 20 | apirr 8459 |
. . . 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 963 wcel 2125 class class class wbr 3961
(class class class)co 5814 cc 7709 cc0 7711 cmul 7716 # cap 8435 |
| 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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-sep 4078 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-cnex 7802 ax-resscn 7803 ax-1cn 7804 ax-1re 7805 ax-icn 7806 ax-addcl 7807 ax-addrcl 7808 ax-mulcl 7809 ax-mulrcl 7810 ax-addcom 7811 ax-mulcom 7812 ax-addass 7813 ax-mulass 7814 ax-distr 7815 ax-i2m1 7816 ax-0lt1 7817 ax-1rid 7818 ax-0id 7819 ax-rnegex 7820 ax-precex 7821 ax-cnre 7822 ax-pre-ltirr 7823 ax-pre-ltwlin 7824 ax-pre-lttrn 7825 ax-pre-apti 7826 ax-pre-ltadd 7827 ax-pre-mulgt0 7828 ax-pre-mulext 7829 |
| This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-nel 2420 df-ral 2437 df-rex 2438 df-reu 2439 df-rab 2441 df-v 2711 df-sbc 2934 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-br 3962 df-opab 4022 df-id 4248 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-iota 5128 df-fun 5165 df-fv 5171 df-riota 5770 df-ov 5817 df-oprab 5818 df-mpo 5819 df-pnf 7893 df-mnf 7894 df-ltxr 7896 df-sub 8027 df-neg 8028 df-reap 8429 df-ap 8436 |
| This theorem is referenced by: msqge0 8470 mulge0 8473 mulap0b 8508 |
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