Theorem muls0ord 28154

Description: If a surreal product is zero, one of its factors must be zero. (Contributed by Scott Fenton, 16-Apr-2025.)

Hypotheses

Ref	Expression
muls0ord.1	⊢ (𝜑 → 𝐴 ∈ No )
muls0ord.2	⊢ (𝜑 → 𝐵 ∈ No )

Assertion

Ref	Expression
muls0ord	⊢ (𝜑 → ((𝐴 ·s 𝐵) = 0s ↔ (𝐴 = 0s ∨ 𝐵 = 0s )))

Proof of Theorem muls0ord

Step	Hyp	Ref	Expression
1		muls0ord.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ No )
2		muls02 28110	. . . . . . . . . . 11 ⊢ (𝐵 ∈ No → ( 0s ·s 𝐵) = 0s )
3	1, 2	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ( 0s ·s 𝐵) = 0s )
4	3	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ( 0s ·s 𝐵) = 0s )
5	4	eqeq2d 2745	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = ( 0s ·s 𝐵) ↔ (𝐴 ·s 𝐵) = 0s ))
6		muls0ord.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ No )
7	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 𝐴 ∈ No )
8		0sno 27797	. . . . . . . . . 10 ⊢ 0s ∈ No
9	8	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 0s ∈ No )
10	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 𝐵 ∈ No )
11		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 𝐵 ≠ 0s )
12	7, 9, 10, 11	mulscan2d 28148	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = ( 0s ·s 𝐵) ↔ 𝐴 = 0s ))
13	5, 12	bitr3d 281	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = 0s ↔ 𝐴 = 0s ))
14	13	biimpd 229	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = 0s → 𝐴 = 0s ))
15	14	impancom 451	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ·s 𝐵) = 0s ) → (𝐵 ≠ 0s → 𝐴 = 0s ))
16	15	necon1bd 2948	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ·s 𝐵) = 0s ) → (¬ 𝐴 = 0s → 𝐵 = 0s ))
17	16	orrd 863	. . 3 ⊢ ((𝜑 ∧ (𝐴 ·s 𝐵) = 0s ) → (𝐴 = 0s ∨ 𝐵 = 0s ))
18	17	ex 412	. 2 ⊢ (𝜑 → ((𝐴 ·s 𝐵) = 0s → (𝐴 = 0s ∨ 𝐵 = 0s )))
19		oveq1 7363	. . . . 5 ⊢ (𝐴 = 0s → (𝐴 ·s 𝐵) = ( 0s ·s 𝐵))
20	19	eqeq1d 2736	. . . 4 ⊢ (𝐴 = 0s → ((𝐴 ·s 𝐵) = 0s ↔ ( 0s ·s 𝐵) = 0s ))
21	3, 20	syl5ibrcom 247	. . 3 ⊢ (𝜑 → (𝐴 = 0s → (𝐴 ·s 𝐵) = 0s ))
22		muls01 28081	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
23	6, 22	syl 17	. . . 4 ⊢ (𝜑 → (𝐴 ·s 0s ) = 0s )
24		oveq2 7364	. . . . 5 ⊢ (𝐵 = 0s → (𝐴 ·s 𝐵) = (𝐴 ·s 0s ))
25	24	eqeq1d 2736	. . . 4 ⊢ (𝐵 = 0s → ((𝐴 ·s 𝐵) = 0s ↔ (𝐴 ·s 0s ) = 0s ))
26	23, 25	syl5ibrcom 247	. . 3 ⊢ (𝜑 → (𝐵 = 0s → (𝐴 ·s 𝐵) = 0s ))
27	21, 26	jaod 859	. 2 ⊢ (𝜑 → ((𝐴 = 0s ∨ 𝐵 = 0s ) → (𝐴 ·s 𝐵) = 0s ))
28	18, 27	impbid 212	1 ⊢ (𝜑 → ((𝐴 ·s 𝐵) = 0s ↔ (𝐴 = 0s ∨ 𝐵 = 0s )))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  (class class class)co 7356   No csur 27605   0_s c0s 27793   ·_s cmuls 28075

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-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-ot 4587  df-uni 4862  df-int 4901  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-1o 8395  df-2o 8396  df-nadd 8592  df-no 27608  df-slt 27609  df-bday 27610  df-sle 27711  df-sslt 27748  df-scut 27750  df-0s 27795  df-made 27815  df-old 27816  df-left 27818  df-right 27819  df-norec 27908  df-norec2 27919  df-adds 27930  df-negs 27990  df-subs 27991  df-muls 28076

This theorem is referenced by:  mulsne0bd  28155  expsne0  28394