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Mirrors > Home > MPE Home > Th. List > mul02lem2 | Structured version Visualization version GIF version |
Description: Lemma for mul02 11437. 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 11222 | . 2 ⊢ 1 ≠ 0 | |
2 | ax-1cn 11211 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
3 | mul02lem1 11435 | . . . . . . . . 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 2741 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → (1 + 1) = 1) |
6 | 5 | oveq2d 7447 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → ((i · i) + (1 + 1)) = ((i · i) + 1)) |
7 | ax-icn 11212 | . . . . . . . . 9 ⊢ i ∈ ℂ | |
8 | 7, 7 | mulcli 11266 | . . . . . . . 8 ⊢ (i · i) ∈ ℂ |
9 | 8, 2, 2 | addassi 11269 | . . . . . . 7 ⊢ (((i · i) + 1) + 1) = ((i · i) + (1 + 1)) |
10 | ax-i2m1 11221 | . . . . . . . 8 ⊢ ((i · i) + 1) = 0 | |
11 | 10 | oveq1i 7441 | . . . . . . 7 ⊢ (((i · i) + 1) + 1) = (0 + 1) |
12 | 9, 11 | eqtr3i 2765 | . . . . . 6 ⊢ ((i · i) + (1 + 1)) = (0 + 1) |
13 | 00id 11434 | . . . . . . 7 ⊢ (0 + 0) = 0 | |
14 | 10, 13 | eqtr4i 2766 | . . . . . 6 ⊢ ((i · i) + 1) = (0 + 0) |
15 | 6, 12, 14 | 3eqtr3g 2798 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → (0 + 1) = (0 + 0)) |
16 | 1re 11259 | . . . . . 6 ⊢ 1 ∈ ℝ | |
17 | 0re 11261 | . . . . . 6 ⊢ 0 ∈ ℝ | |
18 | readdcan 11433 | . . . . . 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 218 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → 1 = 0) |
21 | 20 | ex 412 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 · 𝐴) ≠ 0 → 1 = 0)) |
22 | 21 | necon1d 2960 | . 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 1537 ∈ wcel 2106 ≠ wne 2938 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 ici 11155 + caddc 11156 · cmul 11158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 |
This theorem is referenced by: mul02 11437 rexmul 13310 mbfmulc2lem 25696 i1fmulc 25753 itg1mulc 25754 reabssgn 43626 stoweidlem34 45990 ztprmneprm 48192 nn0sumshdiglemA 48469 nn0sumshdiglem1 48471 |
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