Theorem muls0ord 28186

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 28142	. . . . . . . . . . 11 ⊢ (𝐵 ∈ No → ( 0s ·s 𝐵) = 0s )
3	1, 2	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ( 0s ·s 𝐵) = 0s )
4	3	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ( 0s ·s 𝐵) = 0s )
5	4	eqeq2d 2737	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = ( 0s ·s 𝐵) ↔ (𝐴 ·s 𝐵) = 0s ))
6		muls0ord.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ No )
7	6	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 𝐴 ∈ No )
8		0sno 27856	. . . . . . . . . 10 ⊢ 0s ∈ No
9	8	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 0s ∈ No )
10	1	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 𝐵 ∈ No )
11		simpr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → 𝐵 ≠ 0s )
12	7, 9, 10, 11	mulscan2d 28180	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = ( 0s ·s 𝐵) ↔ 𝐴 = 0s ))
13	5, 12	bitr3d 280	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = 0s ↔ 𝐴 = 0s ))
14	13	biimpd 228	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ≠ 0s ) → ((𝐴 ·s 𝐵) = 0s → 𝐴 = 0s ))
15	14	impancom 450	. . . . 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 861	. . 3 ⊢ ((𝜑 ∧ (𝐴 ·s 𝐵) = 0s ) → (𝐴 = 0s ∨ 𝐵 = 0s ))
18	17	ex 411	. 2 ⊢ (𝜑 → ((𝐴 ·s 𝐵) = 0s → (𝐴 = 0s ∨ 𝐵 = 0s )))
19		oveq1 7431	. . . . 5 ⊢ (𝐴 = 0s → (𝐴 ·s 𝐵) = ( 0s ·s 𝐵))
20	19	eqeq1d 2728	. . . 4 ⊢ (𝐴 = 0s → ((𝐴 ·s 𝐵) = 0s ↔ ( 0s ·s 𝐵) = 0s ))
21	3, 20	syl5ibrcom 246	. . 3 ⊢ (𝜑 → (𝐴 = 0s → (𝐴 ·s 𝐵) = 0s ))
22		muls01 28113	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 ·s 0s ) = 0s )
23	6, 22	syl 17	. . . 4 ⊢ (𝜑 → (𝐴 ·s 0s ) = 0s )
24		oveq2 7432	. . . . 5 ⊢ (𝐵 = 0s → (𝐴 ·s 𝐵) = (𝐴 ·s 0s ))
25	24	eqeq1d 2728	. . . 4 ⊢ (𝐵 = 0s → ((𝐴 ·s 𝐵) = 0s ↔ (𝐴 ·s 0s ) = 0s ))
26	23, 25	syl5ibrcom 246	. . 3 ⊢ (𝜑 → (𝐵 = 0s → (𝐴 ·s 𝐵) = 0s ))
27	21, 26	jaod 857	. 2 ⊢ (𝜑 → ((𝐴 = 0s ∨ 𝐵 = 0s ) → (𝐴 ·s 𝐵) = 0s ))
28	18, 27	impbid 211	1 ⊢ (𝜑 → ((𝐴 ·s 𝐵) = 0s ↔ (𝐴 = 0s ∨ 𝐵 = 0s )))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  (class class class)co 7424   No csur 27669   0_s c0s 27852   ·_s cmuls 28107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-ot 4642  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-1o 8496  df-2o 8497  df-nadd 8696  df-no 27672  df-slt 27673  df-bday 27674  df-sle 27775  df-sslt 27811  df-scut 27813  df-0s 27854  df-made 27871  df-old 27872  df-left 27874  df-right 27875  df-norec 27952  df-norec2 27963  df-adds 27974  df-negs 28031  df-subs 28032  df-muls 28108

This theorem is referenced by:  mulsne0bd  28187