Theorem muls0ord 28389

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 28345	. . . . . . . . . . 11 ⊢ (𝐵 ∈ No → ( 0s ·s 𝐵) = 0s )
3	1, 2	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ( 0s ·s 𝐵) = 0s )
4	3	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ( 0s ·s 𝐵) = 0s )
5	4	eqeq2d 2773	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = ( 0s ·s 𝐵) ↔ (𝐴 ·s 𝐵) = 0s ))
6		muls0ord.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ No )
7	6	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 𝐴 ∈ No )
8		0no 28013	. . . . . . . . . 10 ⊢ 0s ∈ No
9	8	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 0s ∈ No )
10	1	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 𝐵 ∈ No )
11		simpr 489	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 𝐵 ≠ 0s )
12	7, 9, 10, 11	mulscan2d 28383	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = ( 0s ·s 𝐵) ↔ 𝐴 = 0s ))
13	5, 12	bitr3d 284	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = 0s ↔ 𝐴 = 0s ))
14	13	biimpd 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = 0s → 𝐴 = 0s ))
15	14	impancom 456	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ·s 𝐵) = 0s ) → (𝐵 ≠ 0s → 𝐴 = 0s ))
16	15	necon1bd 2975	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ·s 𝐵) = 0s ) → (¬ 𝐴 = 0s → 𝐵 = 0s ))
17	16	orrd 876	. . 3 ⊢ ((𝜑 ∧ (𝐴 ·s 𝐵) = 0s ) → (𝐴 = 0s ∨ 𝐵 = 0s ))
18	17	ex 417	. 2 ⊢ (𝜑 → ((𝐴 ·s 𝐵) = 0s → (𝐴 = 0s ∨ 𝐵 = 0s )))
19		oveq1 7419	. . . . 5 ⊢ (𝐴 = 0s → (𝐴 ·s 𝐵) = ( 0s ·s 𝐵))
20	19	eqeq1d 2764	. . . 4 ⊢ (𝐴 = 0s → ((𝐴 ·s 𝐵) = 0s ↔ ( 0s ·s 𝐵) = 0s ))
21	3, 20	syl5ibrcom 250	. . 3 ⊢ (𝜑 → (𝐴 = 0s → (𝐴 ·s 𝐵) = 0s ))
22		muls01 28316	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
23	6, 22	syl 18	. . . 4 ⊢ (𝜑 → (𝐴 ·s 0s ) = 0s )
24		oveq2 7420	. . . . 5 ⊢ (𝐵 = 0s → (𝐴 ·s 𝐵) = (𝐴 ·s 0s ))
25	24	eqeq1d 2764	. . . 4 ⊢ (𝐵 = 0s → ((𝐴 ·s 𝐵) = 0s ↔ (𝐴 ·s 0s ) = 0s ))
26	23, 25	syl5ibrcom 250	. . 3 ⊢ (𝜑 → (𝐵 = 0s → (𝐴 ·s 𝐵) = 0s ))
27	21, 26	jaod 872	. 2 ⊢ (𝜑 → ((𝐴 = 0s ∨ 𝐵 = 0s ) → (𝐴 ·s 𝐵) = 0s ))
28	18, 27	impbid 215	1 ⊢ (𝜑 → ((𝐴 ·s 𝐵) = 0s ↔ (𝐴 = 0s ∨ 𝐵 = 0s )))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  (class class class)co 7412   No csur 27815   0_s c0s 28009   ·_s cmuls 28310

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-ot 4597  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-1o 8451  df-2o 8452  df-nadd 8650  df-no 27818  df-lts 27819  df-bday 27820  df-les 27920  df-slts 27962  df-cuts 27964  df-0s 28011  df-made 28031  df-old 28032  df-left 28034  df-right 28035  df-norec 28142  df-norec2 28153  df-adds 28164  df-negs 28225  df-subs 28226  df-muls 28311

This theorem is used by:  mulsne0bd  28390  expsne0  28640