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Mirrors > Home > ILE Home > Th. List > mulap0 | GIF version |
Description: The product of two numbers apart from zero is apart from zero. Lemma 2.15 of [Geuvers], p. 6. (Contributed by Jim Kingdon, 22-Feb-2020.) |
Ref | Expression |
---|---|
mulap0 | ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 · 𝐵) # 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recexap 8541 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 # 0) → ∃𝑥 ∈ ℂ (𝐵 · 𝑥) = 1) | |
2 | 1 | adantl 275 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ∃𝑥 ∈ ℂ (𝐵 · 𝑥) = 1) |
3 | simpllr 524 | . . . 4 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑥 ∈ ℂ ∧ (𝐵 · 𝑥) = 1)) → 𝐴 # 0) | |
4 | simplll 523 | . . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑥 ∈ ℂ ∧ (𝐵 · 𝑥) = 1)) → 𝐴 ∈ ℂ) | |
5 | simplrl 525 | . . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑥 ∈ ℂ ∧ (𝐵 · 𝑥) = 1)) → 𝐵 ∈ ℂ) | |
6 | simprl 521 | . . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑥 ∈ ℂ ∧ (𝐵 · 𝑥) = 1)) → 𝑥 ∈ ℂ) | |
7 | 4, 5, 6 | mulassd 7913 | . . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑥 ∈ ℂ ∧ (𝐵 · 𝑥) = 1)) → ((𝐴 · 𝐵) · 𝑥) = (𝐴 · (𝐵 · 𝑥))) |
8 | simprr 522 | . . . . . 6 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑥 ∈ ℂ ∧ (𝐵 · 𝑥) = 1)) → (𝐵 · 𝑥) = 1) | |
9 | 8 | oveq2d 5852 | . . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑥 ∈ ℂ ∧ (𝐵 · 𝑥) = 1)) → (𝐴 · (𝐵 · 𝑥)) = (𝐴 · 1)) |
10 | 4 | mulid1d 7907 | . . . . 5 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑥 ∈ ℂ ∧ (𝐵 · 𝑥) = 1)) → (𝐴 · 1) = 𝐴) |
11 | 7, 9, 10 | 3eqtrd 2201 | . . . 4 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑥 ∈ ℂ ∧ (𝐵 · 𝑥) = 1)) → ((𝐴 · 𝐵) · 𝑥) = 𝐴) |
12 | 6 | mul02d 8281 | . . . 4 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑥 ∈ ℂ ∧ (𝐵 · 𝑥) = 1)) → (0 · 𝑥) = 0) |
13 | 3, 11, 12 | 3brtr4d 4008 | . . 3 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑥 ∈ ℂ ∧ (𝐵 · 𝑥) = 1)) → ((𝐴 · 𝐵) · 𝑥) # (0 · 𝑥)) |
14 | 4, 5 | mulcld 7910 | . . . 4 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑥 ∈ ℂ ∧ (𝐵 · 𝑥) = 1)) → (𝐴 · 𝐵) ∈ ℂ) |
15 | 0cnd 7883 | . . . 4 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑥 ∈ ℂ ∧ (𝐵 · 𝑥) = 1)) → 0 ∈ ℂ) | |
16 | mulext1 8501 | . . . 4 ⊢ (((𝐴 · 𝐵) ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (((𝐴 · 𝐵) · 𝑥) # (0 · 𝑥) → (𝐴 · 𝐵) # 0)) | |
17 | 14, 15, 6, 16 | syl3anc 1227 | . . 3 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑥 ∈ ℂ ∧ (𝐵 · 𝑥) = 1)) → (((𝐴 · 𝐵) · 𝑥) # (0 · 𝑥) → (𝐴 · 𝐵) # 0)) |
18 | 13, 17 | mpd 13 | . 2 ⊢ ((((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) ∧ (𝑥 ∈ ℂ ∧ (𝐵 · 𝑥) = 1)) → (𝐴 · 𝐵) # 0) |
19 | 2, 18 | rexlimddv 2586 | 1 ⊢ (((𝐴 ∈ ℂ ∧ 𝐴 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 · 𝐵) # 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 ∃wrex 2443 class class class wbr 3976 (class class class)co 5836 ℂcc 7742 0cc0 7744 1c1 7745 · cmul 7749 # cap 8470 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 |
This theorem is referenced by: mulap0b 8543 mulap0i 8544 mulap0d 8546 divmuldivap 8599 divdivdivap 8600 divmuleqap 8604 divadddivap 8614 conjmulap 8616 expcl2lemap 10457 expclzaplem 10469 |
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