Theorem mulsproplem4 28278

Description: Lemma for surreal multiplication. Under the inductive hypothesis, the product of a member of the old set of 𝐴 and a member of the old set of 𝐵 is a surreal number. (Contributed by Scott Fenton, 4-Mar-2025.)

Hypotheses

Ref	Expression
mulsproplem.1	⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
mulsproplem4.1	⊢ (𝜑 → 𝑋 ∈ ( O ‘( bday ‘𝐴)))
mulsproplem4.2	⊢ (𝜑 → 𝑌 ∈ ( O ‘( bday ‘𝐵)))

Assertion

Ref	Expression
mulsproplem4	⊢ (𝜑 → (𝑋 ·s 𝑌) ∈ No )

Distinct variable groups:   𝐴,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐵,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐶,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐷,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐸,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐹,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝑋,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝑌,𝑏,𝑐,𝑑,𝑒,𝑓

Allowed substitution hints:   𝜑(𝑒,𝑓,𝑎,𝑏,𝑐,𝑑)   𝑌(𝑎)

Proof of Theorem mulsproplem4

Step	Hyp	Ref	Expression
1		mulsproplem.1	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
2		mulsproplem4.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ( O ‘( bday ‘𝐴)))
3	2	oldnod 28006	. . 3 ⊢ (𝜑 → 𝑋 ∈ No )
4		mulsproplem4.2	. . . 4 ⊢ (𝜑 → 𝑌 ∈ ( O ‘( bday ‘𝐵)))
5	4	oldnod 28006	. . 3 ⊢ (𝜑 → 𝑌 ∈ No )
6		0no 27968	. . . 4 ⊢ 0s ∈ No
7	6	a1i 11	. . 3 ⊢ (𝜑 → 0s ∈ No )
8		bday0 27970	. . . . . . . . . . . 12 ⊢ ( bday ‘ 0s ) = ∅
9	8, 8	oveq12i 7423	. . . . . . . . . . 11 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = (∅ +no ∅)
10		0elon 6417	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
11		naddrid 8670	. . . . . . . . . . . 12 ⊢ (∅ ∈ On → (∅ +no ∅) = ∅)
12	10, 11	ax-mp 5	. . . . . . . . . . 11 ⊢ (∅ +no ∅) = ∅
13	9, 12	eqtri 2792	. . . . . . . . . 10 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = ∅
14	13, 13	uneq12i 4128	. . . . . . . . 9 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = (∅ ∪ ∅)
15		un0 4358	. . . . . . . . 9 ⊢ (∅ ∪ ∅) = ∅
16	14, 15	eqtri 2792	. . . . . . . 8 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = ∅
17	16, 16	uneq12i 4128	. . . . . . 7 ⊢ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))) = (∅ ∪ ∅)
18	17, 15	eqtri 2792	. . . . . 6 ⊢ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))) = ∅
19	18	uneq2i 4127	. . . . 5 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) = ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ ∅)
20		un0 4358	. . . . 5 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ ∅) = (( bday ‘𝑋) +no ( bday ‘𝑌))
21	19, 20	eqtri 2792	. . . 4 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) = (( bday ‘𝑋) +no ( bday ‘𝑌))
22		oldbdayim 28048	. . . . . . 7 ⊢ (𝑋 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑋) ∈ ( bday ‘𝐴))
23	2, 22	syl 18	. . . . . 6 ⊢ (𝜑 → ( bday ‘𝑋) ∈ ( bday ‘𝐴))
24		oldbdayim 28048	. . . . . . 7 ⊢ (𝑌 ∈ ( O ‘( bday ‘𝐵)) → ( bday ‘𝑌) ∈ ( bday ‘𝐵))
25	4, 24	syl 18	. . . . . 6 ⊢ (𝜑 → ( bday ‘𝑌) ∈ ( bday ‘𝐵))
26		bdayon 27911	. . . . . . 7 ⊢ ( bday ‘𝐴) ∈ On
27		bdayon 27911	. . . . . . 7 ⊢ ( bday ‘𝐵) ∈ On
28		naddel12 8687	. . . . . . 7 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → ((( bday ‘𝑋) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑌) ∈ ( bday ‘𝐵)) → (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
29	26, 27, 28	mp2an 704	. . . . . 6 ⊢ ((( bday ‘𝑋) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑌) ∈ ( bday ‘𝐵)) → (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
30	23, 25, 29	syl2anc 595	. . . . 5 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
31		elun1 4143	. . . . 5 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) → (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
32	30, 31	syl 18	. . . 4 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
33	21, 32	eqeltrid 2873	. . 3 ⊢ (𝜑 → ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
34	1, 3, 5, 7, 7, 7, 7, 33	mulsproplem1 28275	. 2 ⊢ (𝜑 → ((𝑋 ·s 𝑌) ∈ No ∧ (( 0s <s 0s ∧ 0s <s 0s ) → (( 0s ·s 0s ) -s ( 0s ·s 0s )) <s (( 0s ·s 0s ) -s ( 0s ·s 0s )))))
35	34	simpld 499	1 ⊢ (𝜑 → (𝑋 ·s 𝑌) ∈ No )

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085   ∪ cun 3911  ∅c0 4294   class class class wbr 5113  Oncon0 6361  ‘cfv 6537  (class class class)co 7411   +no cnadd 8651   No csur 27770   <s clts 27771   bday cbday 27772   0_s c0s 27964   O cold 27982   -_s csubs 28179   ·_s cmuls 28265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-1o 8453  df-2o 8454  df-nadd 8652  df-no 27773  df-lts 27774  df-bday 27775  df-slts 27917  df-cuts 27919  df-0s 27966  df-made 27986  df-old 27987

This theorem is referenced by:  mulsproplem5  28279  mulsproplem6  28280  mulsproplem7  28281  mulsproplem8  28282  mulsproplem9  28283