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Mirrors > Home > MPE Home > Th. List > mul02lem2 | Structured version Visualization version GIF version |
Description: Lemma for mul02 10807. 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 10595 | . 2 ⊢ 1 ≠ 0 | |
2 | ax-1cn 10584 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
3 | mul02lem1 10805 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) ∧ 1 ∈ ℂ) → 1 = (1 + 1)) | |
4 | 2, 3 | mpan2 690 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → 1 = (1 + 1)) |
5 | 4 | eqcomd 2804 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → (1 + 1) = 1) |
6 | 5 | oveq2d 7151 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → ((i · i) + (1 + 1)) = ((i · i) + 1)) |
7 | ax-icn 10585 | . . . . . . . . 9 ⊢ i ∈ ℂ | |
8 | 7, 7 | mulcli 10637 | . . . . . . . 8 ⊢ (i · i) ∈ ℂ |
9 | 8, 2, 2 | addassi 10640 | . . . . . . 7 ⊢ (((i · i) + 1) + 1) = ((i · i) + (1 + 1)) |
10 | ax-i2m1 10594 | . . . . . . . 8 ⊢ ((i · i) + 1) = 0 | |
11 | 10 | oveq1i 7145 | . . . . . . 7 ⊢ (((i · i) + 1) + 1) = (0 + 1) |
12 | 9, 11 | eqtr3i 2823 | . . . . . 6 ⊢ ((i · i) + (1 + 1)) = (0 + 1) |
13 | 00id 10804 | . . . . . . 7 ⊢ (0 + 0) = 0 | |
14 | 10, 13 | eqtr4i 2824 | . . . . . 6 ⊢ ((i · i) + 1) = (0 + 0) |
15 | 6, 12, 14 | 3eqtr3g 2856 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → (0 + 1) = (0 + 0)) |
16 | 1re 10630 | . . . . . 6 ⊢ 1 ∈ ℝ | |
17 | 0re 10632 | . . . . . 6 ⊢ 0 ∈ ℝ | |
18 | readdcan 10803 | . . . . . 6 ⊢ ((1 ∈ ℝ ∧ 0 ∈ ℝ ∧ 0 ∈ ℝ) → ((0 + 1) = (0 + 0) ↔ 1 = 0)) | |
19 | 16, 17, 17, 18 | mp3an 1458 | . . . . 5 ⊢ ((0 + 1) = (0 + 0) ↔ 1 = 0) |
20 | 15, 19 | sylib 221 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (0 · 𝐴) ≠ 0) → 1 = 0) |
21 | 20 | ex 416 | . . 3 ⊢ (𝐴 ∈ ℝ → ((0 · 𝐴) ≠ 0 → 1 = 0)) |
22 | 21 | necon1d 3009 | . 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 ici 10528 + caddc 10529 · cmul 10531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 |
This theorem is referenced by: mul02 10807 rexmul 12652 mbfmulc2lem 24251 i1fmulc 24307 itg1mulc 24308 reabssgn 40336 stoweidlem34 42676 ztprmneprm 44749 nn0sumshdiglemA 45033 nn0sumshdiglem1 45035 |
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