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Mirrors > Home > MPE Home > Th. List > mul02lem2 | Structured version Visualization version GIF version |
Description: Lemma for mul02 11153. Zero times a real is zero. (Contributed by Scott Fenton, 3-Jan-2013.) |
Ref | Expression |
---|---|
mul02lem2 | ⊢ (𝐴 ∈ ℝ → (0 · 𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1ne0 10940 | . 2 ⊢ 1 ≠ 0 | |
2 | ax-1cn 10929 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
3 | mul02lem1 11151 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) ∧ 1 ∈ ℂ) → 1 = (1 + 1)) | |
4 | 2, 3 | mpan2 688 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → 1 = (1 + 1)) |
5 | 4 | eqcomd 2744 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → (1 + 1) = 1) |
6 | 5 | oveq2d 7291 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → ((i · i) + (1 + 1)) = ((i · i) + 1)) |
7 | ax-icn 10930 | . . . . . . . . 9 ⊢ i ∈ ℂ | |
8 | 7, 7 | mulcli 10982 | . . . . . . . 8 ⊢ (i · i) ∈ ℂ |
9 | 8, 2, 2 | addassi 10985 | . . . . . . 7 ⊢ (((i · i) + 1) + 1) = ((i · i) + (1 + 1)) |
10 | ax-i2m1 10939 | . . . . . . . 8 ⊢ ((i · i) + 1) = 0 | |
11 | 10 | oveq1i 7285 | . . . . . . 7 ⊢ (((i · i) + 1) + 1) = (0 + 1) |
12 | 9, 11 | eqtr3i 2768 | . . . . . 6 ⊢ ((i · i) + (1 + 1)) = (0 + 1) |
13 | 00id 11150 | . . . . . . 7 ⊢ (0 + 0) = 0 | |
14 | 10, 13 | eqtr4i 2769 | . . . . . 6 ⊢ ((i · i) + 1) = (0 + 0) |
15 | 6, 12, 14 | 3eqtr3g 2801 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → (0 + 1) = (0 + 0)) |
16 | 1re 10975 | . . . . . 6 ⊢ 1 ∈ ℝ | |
17 | 0re 10977 | . . . . . 6 ⊢ 0 ∈ ℝ | |
18 | readdcan 11149 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ 0 ∈ ℝ ∧ 0 ∈ ℝ) → ((0 + 1) = (0 + 0) ↔ 1 = 0)) | |
19 | 16, 17, 17, 18 | mp3an 1460 | . . . . 5 ⊢ ((0 + 1) = (0 + 0) ↔ 1 = 0) |
20 | 15, 19 | sylib 217 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → 1 = 0) |
21 | 20 | ex 413 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 · 𝐴) ≠ 0 → 1 = 0)) |
22 | 21 | necon1d 2965 | . 2 ⊢ (𝐴 ∈ ℝ → (1 ≠ 0 → (0 · 𝐴) = 0)) |
23 | 1, 22 | mpi 20 | 1 ⊢ (𝐴 ∈ ℝ → (0 · 𝐴) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 ici 10873 + caddc 10874 · cmul 10876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-ltxr 11014 |
This theorem is referenced by: mul02 11153 rexmul 13005 mbfmulc2lem 24811 i1fmulc 24868 itg1mulc 24869 reabssgn 41244 stoweidlem34 43575 ztprmneprm 45683 nn0sumshdiglemA 45965 nn0sumshdiglem1 45967 |
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