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Mirrors > Home > MPE Home > Th. List > xmulid1 | Structured version Visualization version GIF version |
Description: Extended real version of mulid1 10075. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmulid1 | ⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 1) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 11988 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | 1re 10077 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | rexmul 12139 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 ·e 1) = (𝐴 · 1)) | |
4 | 2, 3 | mpan2 707 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 ·e 1) = (𝐴 · 1)) |
5 | ax-1rid 10044 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | |
6 | 4, 5 | eqtrd 2685 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 ·e 1) = 𝐴) |
7 | 2 | rexri 10135 | . . . . 5 ⊢ 1 ∈ ℝ* |
8 | 0lt1 10588 | . . . . 5 ⊢ 0 < 1 | |
9 | xmulpnf2 12143 | . . . . 5 ⊢ ((1 ∈ ℝ* ∧ 0 < 1) → (+∞ ·e 1) = +∞) | |
10 | 7, 8, 9 | mp2an 708 | . . . 4 ⊢ (+∞ ·e 1) = +∞ |
11 | oveq1 6697 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 ·e 1) = (+∞ ·e 1)) | |
12 | id 22 | . . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞) | |
13 | 10, 11, 12 | 3eqtr4a 2711 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 ·e 1) = 𝐴) |
14 | xmulmnf2 12145 | . . . . 5 ⊢ ((1 ∈ ℝ* ∧ 0 < 1) → (-∞ ·e 1) = -∞) | |
15 | 7, 8, 14 | mp2an 708 | . . . 4 ⊢ (-∞ ·e 1) = -∞ |
16 | oveq1 6697 | . . . 4 ⊢ (𝐴 = -∞ → (𝐴 ·e 1) = (-∞ ·e 1)) | |
17 | id 22 | . . . 4 ⊢ (𝐴 = -∞ → 𝐴 = -∞) | |
18 | 15, 16, 17 | 3eqtr4a 2711 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 ·e 1) = 𝐴) |
19 | 6, 13, 18 | 3jaoi 1431 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 ·e 1) = 𝐴) |
20 | 1, 19 | sylbi 207 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 1) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1053 = wceq 1523 ∈ wcel 2030 class class class wbr 4685 (class class class)co 6690 ℝcr 9973 0cc0 9974 1c1 9975 · cmul 9979 +∞cpnf 10109 -∞cmnf 10110 ℝ*cxr 10111 < clt 10112 ·e cxmu 11983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-xneg 11984 df-xmul 11986 |
This theorem is referenced by: xmulid2 12148 xlemul1 12158 xrsmcmn 19817 nmoi2 22581 xdivrec 29763 omssubadd 30490 |
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