Metamath Proof Explorer


Theorem mulscutlem

Description: Lemma for mulscut . State the theorem with extra DV conditions. (Contributed by Scott Fenton, 7-Mar-2025)

Ref Expression
Hypotheses mulscutlem.1 φ A No
mulscutlem.2 φ B No
Assertion mulscutlem φ A s B No a | p L A q L B a = p s B + s A s q - s p s q b | r R A s R B b = r s B + s A s s - s r s s s A s B A s B s c | t L A u R B c = t s B + s A s u - s t s u d | v R A w L B d = v s B + s A s w - s v s w

Proof

Step Hyp Ref Expression
1 mulscutlem.1 φ A No
2 mulscutlem.2 φ B No
3 mulsprop e No f No g No h No i No j No e s f No g < s h i < s j g s j - s g s i < s h s j - s h s i
4 3 a1d e No f No g No h No i No j No bday e + bday f bday g + bday i bday h + bday j bday g + bday j bday h + bday i bday A + bday B bday 0 s + bday 0 s bday 0 s + bday 0 s bday 0 s + bday 0 s bday 0 s + bday 0 s e s f No g < s h i < s j g s j - s g s i < s h s j - s h s i
5 4 3expa e No f No g No h No i No j No bday e + bday f bday g + bday i bday h + bday j bday g + bday j bday h + bday i bday A + bday B bday 0 s + bday 0 s bday 0 s + bday 0 s bday 0 s + bday 0 s bday 0 s + bday 0 s e s f No g < s h i < s j g s j - s g s i < s h s j - s h s i
6 5 ralrimivva e No f No g No h No i No j No bday e + bday f bday g + bday i bday h + bday j bday g + bday j bday h + bday i bday A + bday B bday 0 s + bday 0 s bday 0 s + bday 0 s bday 0 s + bday 0 s bday 0 s + bday 0 s e s f No g < s h i < s j g s j - s g s i < s h s j - s h s i
7 6 ralrimivva e No f No g No h No i No j No bday e + bday f bday g + bday i bday h + bday j bday g + bday j bday h + bday i bday A + bday B bday 0 s + bday 0 s bday 0 s + bday 0 s bday 0 s + bday 0 s bday 0 s + bday 0 s e s f No g < s h i < s j g s j - s g s i < s h s j - s h s i
8 7 rgen2 e No f No g No h No i No j No bday e + bday f bday g + bday i bday h + bday j bday g + bday j bday h + bday i bday A + bday B bday 0 s + bday 0 s bday 0 s + bday 0 s bday 0 s + bday 0 s bday 0 s + bday 0 s e s f No g < s h i < s j g s j - s g s i < s h s j - s h s i
9 8 a1i A No B No e No f No g No h No i No j No bday e + bday f bday g + bday i bday h + bday j bday g + bday j bday h + bday i bday A + bday B bday 0 s + bday 0 s bday 0 s + bday 0 s bday 0 s + bday 0 s bday 0 s + bday 0 s e s f No g < s h i < s j g s j - s g s i < s h s j - s h s i
10 simpl A No B No A No
11 simpr A No B No B No
12 9 10 11 mulsproplem10 A No B No A s B No a | p L A q L B a = p s B + s A s q - s p s q b | r R A s R B b = r s B + s A s s - s r s s s A s B A s B s c | t L A u R B c = t s B + s A s u - s t s u d | v R A w L B d = v s B + s A s w - s v s w
13 1 2 12 syl2anc φ A s B No a | p L A q L B a = p s B + s A s q - s p s q b | r R A s R B b = r s B + s A s s - s r s s s A s B A s B s c | t L A u R B c = t s B + s A s u - s t s u d | v R A w L B d = v s B + s A s w - s v s w