Theorem muls0ord 28280

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 28236	. . . . . . . . . . 11 ⊢ (𝐵 ∈ No → ( 0s ·s 𝐵) = 0s )
3	1, 2	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ( 0s ·s 𝐵) = 0s )
4	3	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ( 0s ·s 𝐵) = 0s )
5	4	eqeq2d 2775	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = ( 0s ·s 𝐵) ↔ (𝐴 ·s 𝐵) = 0s ))
6		muls0ord.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ No )
7	6	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 𝐴 ∈ No )
8		0no 27904	. . . . . . . . . 10 ⊢ 0s ∈ No
9	8	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 0s ∈ No )
10	1	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 𝐵 ∈ No )
11		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 𝐵 ≠ 0s )
12	7, 9, 10, 11	mulscan2d 28274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = ( 0s ·s 𝐵) ↔ 𝐴 = 0s ))
13	5, 12	bitr3d 283	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = 0s ↔ 𝐴 = 0s ))
14	13	biimpd 231	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = 0s → 𝐴 = 0s ))
15	14	impancom 455	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ·s 𝐵) = 0s ) → (𝐵 ≠ 0s → 𝐴 = 0s ))
16	15	necon1bd 2977	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ·s 𝐵) = 0s ) → (¬ 𝐴 = 0s → 𝐵 = 0s ))
17	16	orrd 874	. . 3 ⊢ ((𝜑 ∧ (𝐴 ·s 𝐵) = 0s ) → (𝐴 = 0s ∨ 𝐵 = 0s ))
18	17	ex 416	. 2 ⊢ (𝜑 → ((𝐴 ·s 𝐵) = 0s → (𝐴 = 0s ∨ 𝐵 = 0s )))
19		oveq1 7405	. . . . 5 ⊢ (𝐴 = 0s → (𝐴 ·s 𝐵) = ( 0s ·s 𝐵))
20	19	eqeq1d 2766	. . . 4 ⊢ (𝐴 = 0s → ((𝐴 ·s 𝐵) = 0s ↔ ( 0s ·s 𝐵) = 0s ))
21	3, 20	syl5ibrcom 249	. . 3 ⊢ (𝜑 → (𝐴 = 0s → (𝐴 ·s 𝐵) = 0s ))
22		muls01 28207	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
23	6, 22	syl 17	. . . 4 ⊢ (𝜑 → (𝐴 ·s 0s ) = 0s )
24		oveq2 7406	. . . . 5 ⊢ (𝐵 = 0s → (𝐴 ·s 𝐵) = (𝐴 ·s 0s ))
25	24	eqeq1d 2766	. . . 4 ⊢ (𝐵 = 0s → ((𝐴 ·s 𝐵) = 0s ↔ (𝐴 ·s 0s ) = 0s ))
26	23, 25	syl5ibrcom 249	. . 3 ⊢ (𝜑 → (𝐵 = 0s → (𝐴 ·s 𝐵) = 0s ))
27	21, 26	jaod 870	. 2 ⊢ (𝜑 → ((𝐴 = 0s ∨ 𝐵 = 0s ) → (𝐴 ·s 𝐵) = 0s ))
28	18, 27	impbid 214	1 ⊢ (𝜑 → ((𝐴 ·s 𝐵) = 0s ↔ (𝐴 = 0s ∨ 𝐵 = 0s )))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1562   ∈ wcel 2144   ≠ wne 2959  (class class class)co 7398   No csur 27706   0_s c0s 27900   ·_s cmuls 28201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-1o 8439  df-2o 8440  df-nadd 8638  df-no 27709  df-lts 27710  df-bday 27711  df-les 27811  df-slts 27853  df-cuts 27855  df-0s 27902  df-made 27922  df-old 27923  df-left 27925  df-right 27926  df-norec 28033  df-norec2 28044  df-adds 28055  df-negs 28116  df-subs 28117  df-muls 28202

This theorem is referenced by:  mulsne0bd  28281  expsne0  28531