Theorem mulsproplem4 27564

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		oldssno 27345	. . . 4 ⊢ ( O ‘( bday ‘𝐴)) ⊆ No
3		mulsproplem4.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ( O ‘( bday ‘𝐴)))
4	2, 3	sselid 3979	. . 3 ⊢ (𝜑 → 𝑋 ∈ No )
5		oldssno 27345	. . . 4 ⊢ ( O ‘( bday ‘𝐵)) ⊆ No
6		mulsproplem4.2	. . . 4 ⊢ (𝜑 → 𝑌 ∈ ( O ‘( bday ‘𝐵)))
7	5, 6	sselid 3979	. . 3 ⊢ (𝜑 → 𝑌 ∈ No )
8		0sno 27316	. . . 4 ⊢ 0s ∈ No
9	8	a1i 11	. . 3 ⊢ (𝜑 → 0s ∈ No )
10		bday0s 27318	. . . . . . . . . . . 12 ⊢ ( bday ‘ 0s ) = ∅
11	10, 10	oveq12i 7417	. . . . . . . . . . 11 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = (∅ +no ∅)
12		0elon 6415	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
13		naddrid 8678	. . . . . . . . . . . 12 ⊢ (∅ ∈ On → (∅ +no ∅) = ∅)
14	12, 13	ax-mp 5	. . . . . . . . . . 11 ⊢ (∅ +no ∅) = ∅
15	11, 14	eqtri 2760	. . . . . . . . . 10 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = ∅
16	15, 15	uneq12i 4160	. . . . . . . . 9 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = (∅ ∪ ∅)
17		un0 4389	. . . . . . . . 9 ⊢ (∅ ∪ ∅) = ∅
18	16, 17	eqtri 2760	. . . . . . . 8 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = ∅
19	18, 18	uneq12i 4160	. . . . . . 7 ⊢ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))) = (∅ ∪ ∅)
20	19, 17	eqtri 2760	. . . . . 6 ⊢ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))) = ∅
21	20	uneq2i 4159	. . . . 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)) ∪ ∅)
22		un0 4389	. . . . 5 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ ∅) = (( bday ‘𝑋) +no ( bday ‘𝑌))
23	21, 22	eqtri 2760	. . . 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))
24		oldbdayim 27372	. . . . . . 7 ⊢ (𝑋 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑋) ∈ ( bday ‘𝐴))
25	3, 24	syl 17	. . . . . 6 ⊢ (𝜑 → ( bday ‘𝑋) ∈ ( bday ‘𝐴))
26		oldbdayim 27372	. . . . . . 7 ⊢ (𝑌 ∈ ( O ‘( bday ‘𝐵)) → ( bday ‘𝑌) ∈ ( bday ‘𝐵))
27	6, 26	syl 17	. . . . . 6 ⊢ (𝜑 → ( bday ‘𝑌) ∈ ( bday ‘𝐵))
28		bdayelon 27267	. . . . . . 7 ⊢ ( bday ‘𝐴) ∈ On
29		bdayelon 27267	. . . . . . 7 ⊢ ( bday ‘𝐵) ∈ On
30		naddel12 8695	. . . . . . 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))))
31	28, 29, 30	mp2an 690	. . . . . 6 ⊢ ((( bday ‘𝑋) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑌) ∈ ( bday ‘𝐵)) → (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
32	25, 27, 31	syl2anc 584	. . . . 5 ⊢ (𝜑 → (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
33		elun1 4175	. . . . 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))))))
34	32, 33	syl 17	. . . 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))))))
35	23, 34	eqeltrid 2837	. . 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))))))
36	1, 4, 7, 9, 9, 9, 9, 35	mulsproplem1 27561	. 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 )))))
37	36	simpld 495	1 ⊢ (𝜑 → (𝑋 ·s 𝑌) ∈ No )

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ∪ cun 3945  ∅c0 4321   class class class wbr 5147  Oncon0 6361  ‘cfv 6540  (class class class)co 7405   +no cnadd 8660   No csur 27132   <s cslt 27133   bday cbday 27134   0_s c0s 27312   O cold 27327   -_s csubs 27484   ·_s cmuls 27551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-1o 8462  df-2o 8463  df-nadd 8661  df-no 27135  df-slt 27136  df-bday 27137  df-sslt 27272  df-scut 27274  df-0s 27314  df-made 27331  df-old 27332

This theorem is referenced by:  mulsproplem5  27565  mulsproplem6  27566  mulsproplem7  27567  mulsproplem8  27568  mulsproplem9  27569