Theorem subsge0d

Description: Non-negative subtraction. (Contributed by Scott Fenton, 26-May-2025)

	Ref	Expression
Hypotheses	subsge0d.1	$$⊢\phi \to A\in \mathrm{No}$$
	subsge0d.2	$$⊢\phi \to B\in \mathrm{No}$$
Assertion	subsge0d	Could not format assertion : No typesetting found for |- ( ph -> ( 0s <_s ( A -s B ) <-> B <_s A ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		subsge0d.1	$$⊢\phi \to A\in \mathrm{No}$$
2		subsge0d.2	$$⊢\phi \to B\in \mathrm{No}$$
3		0sno	Could not format 0s e. No : No typesetting found for |- 0s e. No with typecode |-
4	3	a1i	Could not format ( ph -> 0s e. No ) : No typesetting found for |- ( ph -> 0s e. No ) with typecode |-
5	1 2	subscld	Could not format ( ph -> ( A -s B ) e. No ) : No typesetting found for |- ( ph -> ( A -s B ) e. No ) with typecode |-
6	4 5 2	sleadd1d	Could not format ( ph -> ( 0s <_s ( A -s B ) <-> ( 0s +s B ) <_s ( ( A -s B ) +s B ) ) ) : No typesetting found for |- ( ph -> ( 0s <_s ( A -s B ) <-> ( 0s +s B ) <_s ( ( A -s B ) +s B ) ) ) with typecode |-
7		addslid	Could not format ( B e. No -> ( 0s +s B ) = B ) : No typesetting found for |- ( B e. No -> ( 0s +s B ) = B ) with typecode |-
8	2 7	syl	Could not format ( ph -> ( 0s +s B ) = B ) : No typesetting found for |- ( ph -> ( 0s +s B ) = B ) with typecode |-
9		npcans	Could not format ( ( A e. No /\ B e. No ) -> ( ( A -s B ) +s B ) = A ) : No typesetting found for |- ( ( A e. No /\ B e. No ) -> ( ( A -s B ) +s B ) = A ) with typecode |-
10	1 2 9	syl2anc	Could not format ( ph -> ( ( A -s B ) +s B ) = A ) : No typesetting found for |- ( ph -> ( ( A -s B ) +s B ) = A ) with typecode |-
11	8 10	breq12d	Could not format ( ph -> ( ( 0s +s B ) <_s ( ( A -s B ) +s B ) <-> B <_s A ) ) : No typesetting found for |- ( ph -> ( ( 0s +s B ) <_s ( ( A -s B ) +s B ) <-> B <_s A ) ) with typecode |-
12	6 11	bitrd	Could not format ( ph -> ( 0s <_s ( A -s B ) <-> B <_s A ) ) : No typesetting found for |- ( ph -> ( 0s <_s ( A -s B ) <-> B <_s A ) ) with typecode |-