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Theorem domneq0 13746
Description: In a domain, a product is zero iff it has a zero factor. (Contributed by Mario Carneiro, 28-Mar-2015.)
Hypotheses
Ref Expression
domneq0.b 𝐵 = (Base‘𝑅)
domneq0.t · = (.r𝑅)
domneq0.z 0 = (0g𝑅)
Assertion
Ref Expression
domneq0 ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0𝑌 = 0 )))

Proof of Theorem domneq0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 3simpc 998 . . 3 ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐵𝑌𝐵))
2 domneq0.b . . . . . 6 𝐵 = (Base‘𝑅)
3 domneq0.t . . . . . 6 · = (.r𝑅)
4 domneq0.z . . . . . 6 0 = (0g𝑅)
52, 3, 4isdomn 13743 . . . . 5 (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
65simprbi 275 . . . 4 (𝑅 ∈ Domn → ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 )))
763ad2ant1 1020 . . 3 ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑌𝐵) → ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 )))
8 oveq1 5917 . . . . . 6 (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
98eqeq1d 2202 . . . . 5 (𝑥 = 𝑋 → ((𝑥 · 𝑦) = 0 ↔ (𝑋 · 𝑦) = 0 ))
10 eqeq1 2200 . . . . . 6 (𝑥 = 𝑋 → (𝑥 = 0𝑋 = 0 ))
1110orbi1d 792 . . . . 5 (𝑥 = 𝑋 → ((𝑥 = 0𝑦 = 0 ) ↔ (𝑋 = 0𝑦 = 0 )))
129, 11imbi12d 234 . . . 4 (𝑥 = 𝑋 → (((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 )) ↔ ((𝑋 · 𝑦) = 0 → (𝑋 = 0𝑦 = 0 ))))
13 oveq2 5918 . . . . . 6 (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌))
1413eqeq1d 2202 . . . . 5 (𝑦 = 𝑌 → ((𝑋 · 𝑦) = 0 ↔ (𝑋 · 𝑌) = 0 ))
15 eqeq1 2200 . . . . . 6 (𝑦 = 𝑌 → (𝑦 = 0𝑌 = 0 ))
1615orbi2d 791 . . . . 5 (𝑦 = 𝑌 → ((𝑋 = 0𝑦 = 0 ) ↔ (𝑋 = 0𝑌 = 0 )))
1714, 16imbi12d 234 . . . 4 (𝑦 = 𝑌 → (((𝑋 · 𝑦) = 0 → (𝑋 = 0𝑦 = 0 )) ↔ ((𝑋 · 𝑌) = 0 → (𝑋 = 0𝑌 = 0 ))))
1812, 17rspc2va 2878 . . 3 (((𝑋𝐵𝑌𝐵) ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))) → ((𝑋 · 𝑌) = 0 → (𝑋 = 0𝑌 = 0 )))
191, 7, 18syl2anc 411 . 2 ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 · 𝑌) = 0 → (𝑋 = 0𝑌 = 0 )))
20 domnring 13745 . . . . . 6 (𝑅 ∈ Domn → 𝑅 ∈ Ring)
21203ad2ant1 1020 . . . . 5 ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑌𝐵) → 𝑅 ∈ Ring)
22 simp3 1001 . . . . 5 ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑌𝐵) → 𝑌𝐵)
232, 3, 4ringlz 13517 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑌𝐵) → ( 0 · 𝑌) = 0 )
2421, 22, 23syl2anc 411 . . . 4 ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑌𝐵) → ( 0 · 𝑌) = 0 )
25 oveq1 5917 . . . . 5 (𝑋 = 0 → (𝑋 · 𝑌) = ( 0 · 𝑌))
2625eqeq1d 2202 . . . 4 (𝑋 = 0 → ((𝑋 · 𝑌) = 0 ↔ ( 0 · 𝑌) = 0 ))
2724, 26syl5ibrcom 157 . . 3 ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑌𝐵) → (𝑋 = 0 → (𝑋 · 𝑌) = 0 ))
28 simp2 1000 . . . . 5 ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑌𝐵) → 𝑋𝐵)
292, 3, 4ringrz 13518 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (𝑋 · 0 ) = 0 )
3021, 28, 29syl2anc 411 . . . 4 ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑌𝐵) → (𝑋 · 0 ) = 0 )
31 oveq2 5918 . . . . 5 (𝑌 = 0 → (𝑋 · 𝑌) = (𝑋 · 0 ))
3231eqeq1d 2202 . . . 4 (𝑌 = 0 → ((𝑋 · 𝑌) = 0 ↔ (𝑋 · 0 ) = 0 ))
3330, 32syl5ibrcom 157 . . 3 ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑌𝐵) → (𝑌 = 0 → (𝑋 · 𝑌) = 0 ))
3427, 33jaod 718 . 2 ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 = 0𝑌 = 0 ) → (𝑋 · 𝑌) = 0 ))
3519, 34impbid 129 1 ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0𝑌 = 0 )))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 709  w3a 980   = wceq 1364  wcel 2164  wral 2472  cfv 5246  (class class class)co 5910  Basecbs 12605  .rcmulr 12683  0gc0g 12854  Ringcrg 13470  NzRingcnzr 13653  Domncdomn 13730
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-setind 4565  ax-cnex 7953  ax-resscn 7954  ax-1cn 7955  ax-1re 7956  ax-icn 7957  ax-addcl 7958  ax-addrcl 7959  ax-mulcl 7960  ax-addcom 7962  ax-addass 7964  ax-i2m1 7967  ax-0lt1 7968  ax-0id 7970  ax-rnegex 7971  ax-pre-ltirr 7974  ax-pre-ltadd 7978
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253  df-fv 5254  df-riota 5865  df-ov 5913  df-oprab 5914  df-mpo 5915  df-pnf 8046  df-mnf 8047  df-ltxr 8049  df-inn 8973  df-2 9031  df-3 9032  df-ndx 12608  df-slot 12609  df-base 12611  df-sets 12612  df-plusg 12695  df-mulr 12696  df-0g 12856  df-mgm 12926  df-sgrp 12972  df-mnd 12985  df-grp 13062  df-minusg 13063  df-mgp 13395  df-ring 13472  df-nzr 13654  df-domn 13733
This theorem is referenced by:  domnmuln0  13747  znidomb  14117
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