Theorem mulsproplem2 28161

Description: Lemma for surreal multiplication. Under the inductive hypothesis, the product of a member of the old set of 𝐴 and 𝐵 itself 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))))))
mulsproplem2.1	⊢ (𝜑 → 𝑋 ∈ ( O ‘( bday ‘𝐴)))
mulsproplem2.2	⊢ (𝜑 → 𝐵 ∈ No )

Assertion

Ref	Expression
mulsproplem2	⊢ (𝜑 → (𝑋 ·s 𝐵) ∈ 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

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

Proof of Theorem mulsproplem2

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		oldssno 27918	. . . 4 ⊢ ( O ‘( bday ‘𝐴)) ⊆ No
3		mulsproplem2.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ( O ‘( bday ‘𝐴)))
4	2, 3	sselid 4006	. . 3 ⊢ (𝜑 → 𝑋 ∈ No )
5		mulsproplem2.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ No )
6		0sno 27889	. . . 4 ⊢ 0s ∈ No
7	6	a1i 11	. . 3 ⊢ (𝜑 → 0s ∈ No )
8		bday0s 27891	. . . . . . . . . . . 12 ⊢ ( bday ‘ 0s ) = ∅
9	8, 8	oveq12i 7460	. . . . . . . . . . 11 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = (∅ +no ∅)
10		0elon 6449	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
11		naddrid 8739	. . . . . . . . . . . 12 ⊢ (∅ ∈ On → (∅ +no ∅) = ∅)
12	10, 11	ax-mp 5	. . . . . . . . . . 11 ⊢ (∅ +no ∅) = ∅
13	9, 12	eqtri 2768	. . . . . . . . . 10 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = ∅
14	13, 13	uneq12i 4189	. . . . . . . . 9 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = (∅ ∪ ∅)
15		un0 4417	. . . . . . . . 9 ⊢ (∅ ∪ ∅) = ∅
16	14, 15	eqtri 2768	. . . . . . . 8 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = ∅
17	16, 16	uneq12i 4189	. . . . . . 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 2768	. . . . . 6 ⊢ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))) = ∅
19	18	uneq2i 4188	. . . . 5 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝐵)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) = ((( bday ‘𝑋) +no ( bday ‘𝐵)) ∪ ∅)
20		un0 4417	. . . . 5 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝐵)) ∪ ∅) = (( bday ‘𝑋) +no ( bday ‘𝐵))
21	19, 20	eqtri 2768	. . . 4 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝐵)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) = (( bday ‘𝑋) +no ( bday ‘𝐵))
22		oldbdayim 27945	. . . . . . 7 ⊢ (𝑋 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑋) ∈ ( bday ‘𝐴))
23	3, 22	syl 17	. . . . . 6 ⊢ (𝜑 → ( bday ‘𝑋) ∈ ( bday ‘𝐴))
24		bdayelon 27839	. . . . . . 7 ⊢ ( bday ‘𝑋) ∈ On
25		bdayelon 27839	. . . . . . 7 ⊢ ( bday ‘𝐴) ∈ On
26		bdayelon 27839	. . . . . . 7 ⊢ ( bday ‘𝐵) ∈ On
27		naddel1 8743	. . . . . . 7 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝑋) ∈ ( bday ‘𝐴) ↔ (( bday ‘𝑋) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
28	24, 25, 26, 27	mp3an 1461	. . . . . 6 ⊢ (( bday ‘𝑋) ∈ ( bday ‘𝐴) ↔ (( bday ‘𝑋) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
29	23, 28	sylib 218	. . . . 5 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
30		elun1 4205	. . . . 5 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) → (( bday ‘𝑋) +no ( bday ‘𝐵)) ∈ ((( 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))))))
31	29, 30	syl 17	. . . 4 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘𝐵)) ∈ ((( 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	21, 31	eqeltrid 2848	. . 3 ⊢ (𝜑 → ((( bday ‘𝑋) +no ( bday ‘𝐵)) ∪ (((( 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))))))
33	1, 4, 5, 7, 7, 7, 7, 32	mulsproplem1 28160	. 2 ⊢ (𝜑 → ((𝑋 ·s 𝐵) ∈ No ∧ (( 0s <s 0s ∧ 0s <s 0s ) → (( 0s ·s 0s ) -s ( 0s ·s 0s )) <s (( 0s ·s 0s ) -s ( 0s ·s 0s )))))
34	33	simpld 494	1 ⊢ (𝜑 → (𝑋 ·s 𝐵) ∈ No )

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ∪ cun 3974  ∅c0 4352   class class class wbr 5166  Oncon0 6395  ‘cfv 6573  (class class class)co 7448   +no cnadd 8721   No csur 27702   <s cslt 27703   bday cbday 27704   0_s c0s 27885   O cold 27900   -_s csubs 28070   ·_s cmuls 28150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844  df-scut 27846  df-0s 27887  df-made 27904  df-old 27905

This theorem is referenced by:  mulsproplem5  28164  mulsproplem6  28165  mulsproplem7  28166  mulsproplem8  28167  mulsproplem9  28168  mulsproplem13  28172