trimul0or - Mathbox for Jim Kingdon

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Theorem trimul0or 16845

Description: Real number trichotomy implies that if a product is zero, one of its factors must be zero. (Contributed by Jim Kingdon, 27-May-2026.)

**Assertion**
Ref	Expression
trimul0or	$\/$ $\/$ $\/$

Distinct variable group:

Proof of Theorem trimul0or Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 531	. . . . . . . 8 $\/$ $\/$ $/\$ $/\$
2	1	ad3antrrr 492	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ # $/\$
3		simplrr 538	. . . . . . . 8 $\/$ $\/$ $/\$ $/\$ $/\$ #
4	3	ad2antrr 488	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ # $/\$
5		simpllr 536	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ # $/\$ #
6		simplr 529	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ # $/\$ #
7	2, 4, 5, 6	mulap0d 8932	. . . . . 6 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ # $/\$ #
8		simpr 110	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ # $/\$
9	2, 4	mulcld 8294	. . . . . . . 8 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ # $/\$
10		0cn 8266	. . . . . . . 8
11		apti 8896	. . . . . . . 8 $/\$ #
12	9, 10, 11	sylancl 413	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ # $/\$ #
13	8, 12	mpbid 147	. . . . . 6 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ # $/\$ #
14	7, 13	pm2.21dd 625	. . . . 5 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ # $/\$ $\/$
15	14	ex 115	. . . 4 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ # $\/$
16		simpr 110	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ # #
17	3	adantr 276	. . . . . . . 8 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ #
18		apti 8896	. . . . . . . 8 $/\$ #
19	17, 10, 18	sylancl 413	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ # #
20	16, 19	mpbird 167	. . . . . 6 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ #
21	20	olcd 742	. . . . 5 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ # $\/$
22	21	a1d 22	. . . 4 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ # $\/$
23		breq1 4112	. . . . . . 7 # #
24	23	dcbid 846	. . . . . 6 DECID # DECID #
25		breq2 4113	. . . . . . . . . 10 # #
26	25	dcbid 846	. . . . . . . . 9 DECID # DECID #
27		cndcap 16844	. . . . . . . . . . . 12 $\/$ $\/$ DECID #
28	27	biimpi 120	. . . . . . . . . . 11 $\/$ $\/$ DECID #
29	28	adantr 276	. . . . . . . . . 10 $\/$ $\/$ $/\$ $/\$ DECID #
30	29	r19.21bi 2630	. . . . . . . . 9 $\/$ $\/$ $/\$ $/\$ $/\$ DECID #
31		0cnd 8267	. . . . . . . . 9 $\/$ $\/$ $/\$ $/\$ $/\$
32	26, 30, 31	rspcdva 2926	. . . . . . . 8 $\/$ $\/$ $/\$ $/\$ $/\$ DECID #
33	32	ralrimiva 2615	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ DECID #
34	33	adantr 276	. . . . . 6 $\/$ $\/$ $/\$ $/\$ $/\$ # DECID #
35	24, 34, 3	rspcdva 2926	. . . . 5 $\/$ $\/$ $/\$ $/\$ $/\$ # DECID #
36		exmiddc 844	. . . . 5 DECID # # $\/$ #
37	35, 36	syl 14	. . . 4 $\/$ $\/$ $/\$ $/\$ $/\$ # # $\/$ #
38	15, 22, 37	mpjaodan 806	. . 3 $\/$ $\/$ $/\$ $/\$ $/\$ # $\/$
39		simpr 110	. . . . . 6 $\/$ $\/$ $/\$ $/\$ $/\$ # #
40	1	adantr 276	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ $/\$ #
41		apti 8896	. . . . . . 7 $/\$ #
42	40, 10, 41	sylancl 413	. . . . . 6 $\/$ $\/$ $/\$ $/\$ $/\$ # #
43	39, 42	mpbird 167	. . . . 5 $\/$ $\/$ $/\$ $/\$ $/\$ #
44	43	orcd 741	. . . 4 $\/$ $\/$ $/\$ $/\$ $/\$ # $\/$
45	44	a1d 22	. . 3 $\/$ $\/$ $/\$ $/\$ $/\$ # $\/$
46		breq1 4112	. . . . . 6 # #
47	46	dcbid 846	. . . . 5 DECID # DECID #
48	47, 33, 1	rspcdva 2926	. . . 4 $\/$ $\/$ $/\$ $/\$ DECID #
49		exmiddc 844	. . . 4 DECID # # $\/$ #
50	48, 49	syl 14	. . 3 $\/$ $\/$ $/\$ $/\$ # $\/$ #
51	38, 45, 50	mpjaodan 806	. 2 $\/$ $\/$ $/\$ $/\$ $\/$
52	51	ralrimivva 2624	1 $\/$ $\/$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 716 DECID wdc 842 $\/$ w3o 1004

wceq 1398

wcel 2203

wral 2520 class class class wbr 4109 (class class class)co 6050

cc 8125

cr 8126

cc0 8127

cmul 8132

clt 8308 # cap 8855

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-br 4110 df-opab 4172 df-mpt 4173 df-id 4414 df-po 4417 df-iso 4418 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-2 9296 df-cj 11527 df-re 11528 df-im 11529

This theorem is referenced by: (None)

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