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Mirrors > Home > MPE Home > Th. List > ax-1rid | Structured version Visualization version GIF version |
Description: 1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by theorem ax1rid 10583. Weakened from the original axiom in the form of statement in mulid1 10639, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.) |
Ref | Expression |
---|---|
ax-1rid | ⊢ (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | cr 10536 | . . 3 class ℝ | |
3 | 1, 2 | wcel 2114 | . 2 wff 𝐴 ∈ ℝ |
4 | c1 10538 | . . . 4 class 1 | |
5 | cmul 10542 | . . . 4 class · | |
6 | 1, 4, 5 | co 7156 | . . 3 class (𝐴 · 1) |
7 | 6, 1 | wceq 1537 | . 2 wff (𝐴 · 1) = 𝐴 |
8 | 3, 7 | wi 4 | 1 wff (𝐴 ∈ ℝ → (𝐴 · 1) = 𝐴) |
Colors of variables: wff setvar class |
This axiom is referenced by: mulid1 10639 mulgt1 11499 ltmulgt11 11500 lemulge11 11502 nnmulcl 11662 addltmul 11874 xmulid1 12673 2submod 13301 cshw1 14184 bezoutlem1 15887 cshwshashnsame 16437 numclwlk1lem1 28148 numclwwlk6 28169 nmopub2tALT 29686 nmfnleub2 29703 unitdivcld 31144 zrhre 31260 sgnmulrp2 31801 knoppcnlem4 33835 1t1e1ALT 39175 remulcan2d 39176 sn-1ne2 39178 nnadddir 39183 nnmul1com 39184 sn-00idlem1 39248 sn-00idlem3 39250 remul02 39255 remul01 39257 sn-0lt1 39266 remulinvcom 39268 remulid2 39269 3cubeslem1 39301 relexpmulnn 40074 relogbmulbexp 44641 line2xlem 44760 line2x 44761 |
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