Metamath Proof Explorer


Theorem rrh0

Description: The image of 0 by the RRHom homomorphism is the ring's zero. (Contributed by Thierry Arnoux, 22-Oct-2017)

Ref Expression
Assertion rrh0 RℝExtℝHomR0=0R

Proof

Step Hyp Ref Expression
1 zssq
2 0z 0
3 1 2 sselii 0
4 simpl RℝExt0RℝExt
5 simpr RℝExt00
6 rrhqima RℝExt0ℝHomR0=ℚHomR0
7 4 5 6 syl2anc RℝExt0ℝHomR0=ℚHomR0
8 3 7 mpan2 RℝExtℝHomR0=ℚHomR0
9 rrextdrg RℝExtRDivRing
10 rrextchr RℝExtchrR=0
11 eqid BaseR=BaseR
12 eqid /rR=/rR
13 eqid ℤRHomR=ℤRHomR
14 11 12 13 qqh0 RDivRingchrR=0ℚHomR0=0R
15 9 10 14 syl2anc RℝExtℚHomR0=0R
16 8 15 eqtrd RℝExtℝHomR0=0R