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Mirrors > Home > MPE Home > Th. List > xmulid1 | Structured version Visualization version GIF version |
Description: Extended real version of mulid1 10633. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmulid1 | ⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 1) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 12505 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | 1re 10635 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | rexmul 12658 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (𝐴 ·e 1) = (𝐴 · 1)) | |
4 | 2, 3 | mpan2 689 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 ·e 1) = (𝐴 · 1)) |
5 | ax-1rid 10601 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | |
6 | 4, 5 | eqtrd 2856 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 ·e 1) = 𝐴) |
7 | 1xr 10694 | . . . . 5 ⊢ 1 ∈ ℝ* | |
8 | 0lt1 11156 | . . . . 5 ⊢ 0 < 1 | |
9 | xmulpnf2 12662 | . . . . 5 ⊢ ((1 ∈ ℝ* ∧ 0 < 1) → (+∞ ·e 1) = +∞) | |
10 | 7, 8, 9 | mp2an 690 | . . . 4 ⊢ (+∞ ·e 1) = +∞ |
11 | oveq1 7157 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 ·e 1) = (+∞ ·e 1)) | |
12 | id 22 | . . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞) | |
13 | 10, 11, 12 | 3eqtr4a 2882 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 ·e 1) = 𝐴) |
14 | xmulmnf2 12664 | . . . . 5 ⊢ ((1 ∈ ℝ* ∧ 0 < 1) → (-∞ ·e 1) = -∞) | |
15 | 7, 8, 14 | mp2an 690 | . . . 4 ⊢ (-∞ ·e 1) = -∞ |
16 | oveq1 7157 | . . . 4 ⊢ (𝐴 = -∞ → (𝐴 ·e 1) = (-∞ ·e 1)) | |
17 | id 22 | . . . 4 ⊢ (𝐴 = -∞ → 𝐴 = -∞) | |
18 | 15, 16, 17 | 3eqtr4a 2882 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 ·e 1) = 𝐴) |
19 | 6, 13, 18 | 3jaoi 1423 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 ·e 1) = 𝐴) |
20 | 1, 19 | sylbi 219 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 1) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1082 = wceq 1533 ∈ wcel 2110 class class class wbr 5059 (class class class)co 7150 ℝcr 10530 0cc0 10531 1c1 10532 · cmul 10536 +∞cpnf 10666 -∞cmnf 10667 ℝ*cxr 10668 < clt 10669 ·e cxmu 12500 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-xneg 12501 df-xmul 12503 |
This theorem is referenced by: xmulid2 12667 xlemul1 12677 xrsmcmn 20562 nmoi2 23333 xdivrec 30598 omssubadd 31553 |
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