![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > remulid2 | Structured version Visualization version GIF version |
Description: Commuted version of ax-1rid 11079 without ax-mulcom 11073. (Contributed by SN, 5-Feb-2024.) |
Ref | Expression |
---|---|
remulid2 | ⊢ (𝐴 ∈ ℝ → (1 · 𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ne 2942 | . . 3 ⊢ (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0) | |
2 | ax-rrecex 11081 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → ∃𝑥 ∈ ℝ (𝐴 · 𝑥) = 1) | |
3 | simpll 765 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → 𝐴 ∈ ℝ) | |
4 | 3 | recnd 11141 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → 𝐴 ∈ ℂ) |
5 | simprl 769 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → 𝑥 ∈ ℝ) | |
6 | 5 | recnd 11141 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → 𝑥 ∈ ℂ) |
7 | 4, 6, 4 | mulassd 11136 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → ((𝐴 · 𝑥) · 𝐴) = (𝐴 · (𝑥 · 𝐴))) |
8 | simprr 771 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → (𝐴 · 𝑥) = 1) | |
9 | 8 | oveq1d 7366 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → ((𝐴 · 𝑥) · 𝐴) = (1 · 𝐴)) |
10 | 3, 5, 8 | remulinvcom 40810 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → (𝑥 · 𝐴) = 1) |
11 | 10 | oveq2d 7367 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → (𝐴 · (𝑥 · 𝐴)) = (𝐴 · 1)) |
12 | ax-1rid 11079 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) | |
13 | 3, 12 | syl 17 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → (𝐴 · 1) = 𝐴) |
14 | 11, 13 | eqtrd 2777 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → (𝐴 · (𝑥 · 𝐴)) = 𝐴) |
15 | 7, 9, 14 | 3eqtr3d 2785 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) ∧ (𝑥 ∈ ℝ ∧ (𝐴 · 𝑥) = 1)) → (1 · 𝐴) = 𝐴) |
16 | 2, 15 | rexlimddv 3156 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (1 · 𝐴) = 𝐴) |
17 | 16 | ex 413 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 ≠ 0 → (1 · 𝐴) = 𝐴)) |
18 | 1, 17 | biimtrrid 242 | . 2 ⊢ (𝐴 ∈ ℝ → (¬ 𝐴 = 0 → (1 · 𝐴) = 𝐴)) |
19 | 1re 11113 | . . . 4 ⊢ 1 ∈ ℝ | |
20 | remul01 40785 | . . . 4 ⊢ (1 ∈ ℝ → (1 · 0) = 0) | |
21 | 19, 20 | mp1i 13 | . . 3 ⊢ (𝐴 = 0 → (1 · 0) = 0) |
22 | oveq2 7359 | . . 3 ⊢ (𝐴 = 0 → (1 · 𝐴) = (1 · 0)) | |
23 | id 22 | . . 3 ⊢ (𝐴 = 0 → 𝐴 = 0) | |
24 | 21, 22, 23 | 3eqtr4d 2787 | . 2 ⊢ (𝐴 = 0 → (1 · 𝐴) = 𝐴) |
25 | 18, 24 | pm2.61d2 181 | 1 ⊢ (𝐴 ∈ ℝ → (1 · 𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 (class class class)co 7351 ℝcr 11008 0cc0 11009 1c1 11010 · cmul 11014 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-ltxr 11152 df-2 12174 df-3 12175 df-resub 40744 |
This theorem is referenced by: sn-mulid2 40813 remulcand 40816 sn-0tie0 40817 |
Copyright terms: Public domain | W3C validator |