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Mirrors > Home > MPE Home > Th. List > mul02lem2 | Structured version Visualization version GIF version |
Description: Lemma for mul02 10416. 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 10207 | . 2 ⊢ 1 ≠ 0 | |
2 | ax-1cn 10196 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
3 | mul02lem1 10414 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) ∧ 1 ∈ ℂ) → 1 = (1 + 1)) | |
4 | 2, 3 | mpan2 671 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → 1 = (1 + 1)) |
5 | 4 | eqcomd 2777 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → (1 + 1) = 1) |
6 | 5 | oveq2d 6809 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → ((i · i) + (1 + 1)) = ((i · i) + 1)) |
7 | ax-icn 10197 | . . . . . . . . 9 ⊢ i ∈ ℂ | |
8 | 7, 7 | mulcli 10247 | . . . . . . . 8 ⊢ (i · i) ∈ ℂ |
9 | 8, 2, 2 | addassi 10250 | . . . . . . 7 ⊢ (((i · i) + 1) + 1) = ((i · i) + (1 + 1)) |
10 | ax-i2m1 10206 | . . . . . . . 8 ⊢ ((i · i) + 1) = 0 | |
11 | 10 | oveq1i 6803 | . . . . . . 7 ⊢ (((i · i) + 1) + 1) = (0 + 1) |
12 | 9, 11 | eqtr3i 2795 | . . . . . 6 ⊢ ((i · i) + (1 + 1)) = (0 + 1) |
13 | 00id 10413 | . . . . . . 7 ⊢ (0 + 0) = 0 | |
14 | 10, 13 | eqtr4i 2796 | . . . . . 6 ⊢ ((i · i) + 1) = (0 + 0) |
15 | 6, 12, 14 | 3eqtr3g 2828 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → (0 + 1) = (0 + 0)) |
16 | 1re 10241 | . . . . . 6 ⊢ 1 ∈ ℝ | |
17 | 0re 10242 | . . . . . 6 ⊢ 0 ∈ ℝ | |
18 | readdcan 10412 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ 0 ∈ ℝ ∧ 0 ∈ ℝ) → ((0 + 1) = (0 + 0) ↔ 1 = 0)) | |
19 | 16, 17, 17, 18 | mp3an 1572 | . . . . 5 ⊢ ((0 + 1) = (0 + 0) ↔ 1 = 0) |
20 | 15, 19 | sylib 208 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → 1 = 0) |
21 | 20 | ex 397 | . . 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 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 (class class class)co 6793 ℂcc 10136 ℝcr 10137 0cc0 10138 1c1 10139 ici 10140 + caddc 10141 · cmul 10143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6796 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-ltxr 10281 |
This theorem is referenced by: mul02 10416 rexmul 12306 mbfmulc2lem 23634 i1fmulc 23690 itg1mulc 23691 stoweidlem34 40768 ztprmneprm 42653 nn0sumshdiglemA 42941 nn0sumshdiglem1 42943 |
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