Theorem ltadd12dd

Description: Addition to both sides of 'less than'. (Contributed by Glauco Siliprandi, 11-Oct-2020)

	Ref	Expression
Hypotheses	ltadd12dd.a	|- ( ph -> A e. RR )
	ltadd12dd.b	|- ( ph -> B e. RR )
	ltadd12dd.c	|- ( ph -> C e. RR )
	ltadd12dd.d	|- ( ph -> D e. RR )
	ltadd12dd.ac	|- ( ph -> A < C )
	ltadd12dd.bd	|- ( ph -> B < D )
Assertion	ltadd12dd	|- ( ph -> ( A + B ) < ( C + D ) )

Proof

Step	Hyp	Ref	Expression
1		ltadd12dd.a	|- ( ph -> A e. RR )
2		ltadd12dd.b	|- ( ph -> B e. RR )
3		ltadd12dd.c	|- ( ph -> C e. RR )
4		ltadd12dd.d	|- ( ph -> D e. RR )
5		ltadd12dd.ac	|- ( ph -> A < C )
6		ltadd12dd.bd	|- ( ph -> B < D )
7	1 2	readdcld	|- ( ph -> ( A + B ) e. RR )
8	3 2	readdcld	|- ( ph -> ( C + B ) e. RR )
9	3 4	readdcld	|- ( ph -> ( C + D ) e. RR )
10	1 3 2 5	ltadd1dd	|- ( ph -> ( A + B ) < ( C + B ) )
11	2 4 3 6	ltadd2dd	|- ( ph -> ( C + B ) < ( C + D ) )
12	7 8 9 10 11	lttrd	|- ( ph -> ( A + B ) < ( C + D ) )