trimul0or - Mathbox for Jim Kingdon

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Theorem trimul0or 16971

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 8949	. . . . . 6 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ # $/\$ #
8		simpr 110	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ # $/\$
9	2, 4	mulcld 8310	. . . . . . . 8 $\/$ $\/$ $/\$ $/\$ $/\$ # $/\$ # $/\$
10		0cn 8282	. . . . . . . 8
11		apti 8913	. . . . . . . 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 8913	. . . . . . . 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 4117	. . . . . . 7 # #
24	23	dcbid 846	. . . . . 6 DECID # DECID #
25		breq2 4118	. . . . . . . . . 10 # #
26	25	dcbid 846	. . . . . . . . 9 DECID # DECID #
27		cndcap 16970	. . . . . . . . . . . 12 $\/$ $\/$ DECID #
28	27	biimpi 120	. . . . . . . . . . 11 $\/$ $\/$ DECID #
29	28	adantr 276	. . . . . . . . . 10 $\/$ $\/$ $/\$ $/\$ DECID #
30	29	r19.21bi 2632	. . . . . . . . 9 $\/$ $\/$ $/\$ $/\$ $/\$ DECID #
31		0cnd 8283	. . . . . . . . 9 $\/$ $\/$ $/\$ $/\$ $/\$
32	26, 30, 31	rspcdva 2928	. . . . . . . 8 $\/$ $\/$ $/\$ $/\$ $/\$ DECID #
33	32	ralrimiva 2617	. . . . . . 7 $\/$ $\/$ $/\$ $/\$ DECID #
34	33	adantr 276	. . . . . 6 $\/$ $\/$ $/\$ $/\$ $/\$ # DECID #
35	24, 34, 3	rspcdva 2928	. . . . 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 8913	. . . . . . 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 4117	. . . . . 6 # #
47	46	dcbid 846	. . . . 5 DECID # DECID #
48	47, 33, 1	rspcdva 2928	. . . 4 $\/$ $\/$ $/\$ $/\$ DECID #
49		exmiddc 844	. . . 4 DECID # # $\/$ #
50	48, 49	syl 14	. . 3 $\/$ $\/$ $/\$ $/\$ # $\/$ #
51	38, 45, 50	mpjaodan 806	. 2 $\/$ $\/$ $/\$ $/\$ $\/$
52	51	ralrimivva 2626	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 2205

wral 2522 class class class wbr 4114 (class class class)co 6058

cc 8141

cr 8142

cc0 8143

cmul 8148

clt 8324 # cap 8872

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 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261

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 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-br 4115 df-opab 4177 df-mpt 4178 df-id 4419 df-po 4422 df-iso 4423 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-2 9313 df-cj 11552 df-re 11553 df-im 11554

This theorem is referenced by: (None)

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