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| Description: Lemma for mul02 11440. 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 11225 | . 2 ⊢ 1 ≠ 0 | |
| 2 | ax-1cn 11214 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
| 3 | mul02lem1 11438 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) ∧ 1 ∈ ℂ) → 1 = (1 + 1)) | |
| 4 | 2, 3 | mpan2 691 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → 1 = (1 + 1)) | 
| 5 | 4 | eqcomd 2742 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → (1 + 1) = 1) | 
| 6 | 5 | oveq2d 7448 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → ((i · i) + (1 + 1)) = ((i · i) + 1)) | 
| 7 | ax-icn 11215 | . . . . . . . . 9 ⊢ i ∈ ℂ | |
| 8 | 7, 7 | mulcli 11269 | . . . . . . . 8 ⊢ (i · i) ∈ ℂ | 
| 9 | 8, 2, 2 | addassi 11272 | . . . . . . 7 ⊢ (((i · i) + 1) + 1) = ((i · i) + (1 + 1)) | 
| 10 | ax-i2m1 11224 | . . . . . . . 8 ⊢ ((i · i) + 1) = 0 | |
| 11 | 10 | oveq1i 7442 | . . . . . . 7 ⊢ (((i · i) + 1) + 1) = (0 + 1) | 
| 12 | 9, 11 | eqtr3i 2766 | . . . . . 6 ⊢ ((i · i) + (1 + 1)) = (0 + 1) | 
| 13 | 00id 11437 | . . . . . . 7 ⊢ (0 + 0) = 0 | |
| 14 | 10, 13 | eqtr4i 2767 | . . . . . 6 ⊢ ((i · i) + 1) = (0 + 0) | 
| 15 | 6, 12, 14 | 3eqtr3g 2799 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → (0 + 1) = (0 + 0)) | 
| 16 | 1re 11262 | . . . . . 6 ⊢ 1 ∈ ℝ | |
| 17 | 0re 11264 | . . . . . 6 ⊢ 0 ∈ ℝ | |
| 18 | readdcan 11436 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ 0 ∈ ℝ ∧ 0 ∈ ℝ) → ((0 + 1) = (0 + 0) ↔ 1 = 0)) | |
| 19 | 16, 17, 17, 18 | mp3an 1462 | . . . . 5 ⊢ ((0 + 1) = (0 + 0) ↔ 1 = 0) | 
| 20 | 15, 19 | sylib 218 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → 1 = 0) | 
| 21 | 20 | ex 412 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 · 𝐴) ≠ 0 → 1 = 0)) | 
| 22 | 21 | necon1d 2961 | . 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 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 (class class class)co 7432 ℂcc 11154 ℝcr 11155 0cc0 11156 1c1 11157 ici 11158 + caddc 11159 · cmul 11161 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-ltxr 11301 | 
| This theorem is referenced by: mul02 11440 rexmul 13314 mbfmulc2lem 25683 i1fmulc 25739 itg1mulc 25740 reabssgn 43654 stoweidlem34 46054 ztprmneprm 48268 nn0sumshdiglemA 48545 nn0sumshdiglem1 48547 | 
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