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Mirrors > Home > MPE Home > Th. List > mul02lem2 | Structured version Visualization version GIF version |
Description: Lemma for mul02 11389. 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 11175 | . 2 ⊢ 1 ≠ 0 | |
2 | ax-1cn 11164 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
3 | mul02lem1 11387 | . . . . . . . . 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 2730 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → (1 + 1) = 1) |
6 | 5 | oveq2d 7417 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → ((i · i) + (1 + 1)) = ((i · i) + 1)) |
7 | ax-icn 11165 | . . . . . . . . 9 ⊢ i ∈ ℂ | |
8 | 7, 7 | mulcli 11218 | . . . . . . . 8 ⊢ (i · i) ∈ ℂ |
9 | 8, 2, 2 | addassi 11221 | . . . . . . 7 ⊢ (((i · i) + 1) + 1) = ((i · i) + (1 + 1)) |
10 | ax-i2m1 11174 | . . . . . . . 8 ⊢ ((i · i) + 1) = 0 | |
11 | 10 | oveq1i 7411 | . . . . . . 7 ⊢ (((i · i) + 1) + 1) = (0 + 1) |
12 | 9, 11 | eqtr3i 2754 | . . . . . 6 ⊢ ((i · i) + (1 + 1)) = (0 + 1) |
13 | 00id 11386 | . . . . . . 7 ⊢ (0 + 0) = 0 | |
14 | 10, 13 | eqtr4i 2755 | . . . . . 6 ⊢ ((i · i) + 1) = (0 + 0) |
15 | 6, 12, 14 | 3eqtr3g 2787 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → (0 + 1) = (0 + 0)) |
16 | 1re 11211 | . . . . . 6 ⊢ 1 ∈ ℝ | |
17 | 0re 11213 | . . . . . 6 ⊢ 0 ∈ ℝ | |
18 | readdcan 11385 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ 0 ∈ ℝ ∧ 0 ∈ ℝ) → ((0 + 1) = (0 + 0) ↔ 1 = 0)) | |
19 | 16, 17, 17, 18 | mp3an 1457 | . . . . 5 ⊢ ((0 + 1) = (0 + 0) ↔ 1 = 0) |
20 | 15, 19 | sylib 217 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → 1 = 0) |
21 | 20 | ex 412 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 · 𝐴) ≠ 0 → 1 = 0)) |
22 | 21 | necon1d 2954 | . 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 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 (class class class)co 7401 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 ici 11108 + caddc 11109 · cmul 11111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-ltxr 11250 |
This theorem is referenced by: mul02 11389 rexmul 13247 mbfmulc2lem 25498 i1fmulc 25555 itg1mulc 25556 reabssgn 42876 stoweidlem34 45235 ztprmneprm 47212 nn0sumshdiglemA 47493 nn0sumshdiglem1 47495 |
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