Theorem 2arwcatlem5 49865

Description: Lemma for 2arwcat 49866. (Contributed by Zhi Wang, 5-Nov-2025.)

Hypotheses

Ref	Expression
2arwcatlem5.1	⊢ (𝜑 → ( 1 · 0 ) = 0 )
2arwcatlem5.2	⊢ (𝜑 → ( 0 · 1 ) = 0 )
2arwcatlem5.3	⊢ (𝜑 → ( 0 · 0 ) ∈ { 0 , 1 })

Assertion

Ref	Expression
2arwcatlem5	⊢ (𝜑 → (( 0 · 0 ) · 0 ) = ( 0 · ( 0 · 0 )))

Proof of Theorem 2arwcatlem5

Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ( 0 · 0 ) = 0 ) → ( 0 · 0 ) = 0 )
2	1	oveq1d 7373	. . 3 ⊢ ((𝜑 ∧ ( 0 · 0 ) = 0 ) → (( 0 · 0 ) · 0 ) = ( 0 · 0 ))
3	1	oveq2d 7374	. . 3 ⊢ ((𝜑 ∧ ( 0 · 0 ) = 0 ) → ( 0 · ( 0 · 0 )) = ( 0 · 0 ))
4	2, 3	eqtr4d 2774	. 2 ⊢ ((𝜑 ∧ ( 0 · 0 ) = 0 ) → (( 0 · 0 ) · 0 ) = ( 0 · ( 0 · 0 )))
5		2arwcatlem5.1	. . . . 5 ⊢ (𝜑 → ( 1 · 0 ) = 0 )
6		2arwcatlem5.2	. . . . 5 ⊢ (𝜑 → ( 0 · 1 ) = 0 )
7	5, 6	eqtr4d 2774	. . . 4 ⊢ (𝜑 → ( 1 · 0 ) = ( 0 · 1 ))
8	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ ( 0 · 0 ) = 1 ) → ( 1 · 0 ) = ( 0 · 1 ))
9		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ( 0 · 0 ) = 1 ) → ( 0 · 0 ) = 1 )
10	9	oveq1d 7373	. . 3 ⊢ ((𝜑 ∧ ( 0 · 0 ) = 1 ) → (( 0 · 0 ) · 0 ) = ( 1 · 0 ))
11	9	oveq2d 7374	. . 3 ⊢ ((𝜑 ∧ ( 0 · 0 ) = 1 ) → ( 0 · ( 0 · 0 )) = ( 0 · 1 ))
12	8, 10, 11	3eqtr4d 2781	. 2 ⊢ ((𝜑 ∧ ( 0 · 0 ) = 1 ) → (( 0 · 0 ) · 0 ) = ( 0 · ( 0 · 0 )))
13		2arwcatlem5.3	. . 3 ⊢ (𝜑 → ( 0 · 0 ) ∈ { 0 , 1 })
14		ovex 7391	. . . 4 ⊢ ( 0 · 0 ) ∈ V
15	14	elpr 4605	. . 3 ⊢ (( 0 · 0 ) ∈ { 0 , 1 } ↔ (( 0 · 0 ) = 0 ∨ ( 0 · 0 ) = 1 ))
16	13, 15	sylib 218	. 2 ⊢ (𝜑 → (( 0 · 0 ) = 0 ∨ ( 0 · 0 ) = 1 ))
17	4, 12, 16	mpjaodan 960	1 ⊢ (𝜑 → (( 0 · 0 ) · 0 ) = ( 0 · ( 0 · 0 )))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  {cpr 4582  (class class class)co 7358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-nul 5251

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361

This theorem is referenced by:  2arwcat  49866