Theorem mulsproplem3 28077

Description: Lemma for surreal multiplication. Under the inductive hypothesis, the product of 𝐴 itself 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))))))
mulsproplem3.1	⊢ (𝜑 → 𝐴 ∈ No )
mulsproplem3.2	⊢ (𝜑 → 𝑌 ∈ ( O ‘( bday ‘𝐵)))

Assertion

Ref	Expression
mulsproplem3	⊢ (𝜑 → (𝐴 ·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   𝑌,𝑏,𝑐,𝑑,𝑒,𝑓

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

Proof of Theorem mulsproplem3

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		mulsproplem3.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
3		oldssno 27822	. . . 4 ⊢ ( O ‘( bday ‘𝐵)) ⊆ No
4		mulsproplem3.2	. . . 4 ⊢ (𝜑 → 𝑌 ∈ ( O ‘( bday ‘𝐵)))
5	3, 4	sselid 3928	. . 3 ⊢ (𝜑 → 𝑌 ∈ No )
6		0sno 27790	. . . 4 ⊢ 0s ∈ No
7	6	a1i 11	. . 3 ⊢ (𝜑 → 0s ∈ No )
8		bday0s 27792	. . . . . . . . . . . 12 ⊢ ( bday ‘ 0s ) = ∅
9	8, 8	oveq12i 7367	. . . . . . . . . . 11 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = (∅ +no ∅)
10		0elon 6369	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
11		naddrid 8607	. . . . . . . . . . . 12 ⊢ (∅ ∈ On → (∅ +no ∅) = ∅)
12	10, 11	ax-mp 5	. . . . . . . . . . 11 ⊢ (∅ +no ∅) = ∅
13	9, 12	eqtri 2756	. . . . . . . . . 10 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = ∅
14	13, 13	uneq12i 4115	. . . . . . . . 9 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = (∅ ∪ ∅)
15		un0 4343	. . . . . . . . 9 ⊢ (∅ ∪ ∅) = ∅
16	14, 15	eqtri 2756	. . . . . . . 8 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = ∅
17	16, 16	uneq12i 4115	. . . . . . 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 2756	. . . . . 6 ⊢ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))) = ∅
19	18	uneq2i 4114	. . . . 5 ⊢ ((( bday ‘𝐴) +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 ‘𝑌)) ∪ ∅)
20		un0 4343	. . . . 5 ⊢ ((( bday ‘𝐴) +no ( bday ‘𝑌)) ∪ ∅) = (( bday ‘𝐴) +no ( bday ‘𝑌))
21	19, 20	eqtri 2756	. . . 4 ⊢ ((( bday ‘𝐴) +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 ‘𝑌))
22		oldbdayim 27854	. . . . . . 7 ⊢ (𝑌 ∈ ( O ‘( bday ‘𝐵)) → ( bday ‘𝑌) ∈ ( bday ‘𝐵))
23	4, 22	syl 17	. . . . . 6 ⊢ (𝜑 → ( bday ‘𝑌) ∈ ( bday ‘𝐵))
24		bdayelon 27735	. . . . . . 7 ⊢ ( bday ‘𝑌) ∈ On
25		bdayelon 27735	. . . . . . 7 ⊢ ( bday ‘𝐵) ∈ On
26		bdayelon 27735	. . . . . . 7 ⊢ ( bday ‘𝐴) ∈ On
27		naddel2 8612	. . . . . . 7 ⊢ ((( bday ‘𝑌) ∈ On ∧ ( bday ‘𝐵) ∈ On ∧ ( bday ‘𝐴) ∈ On) → (( bday ‘𝑌) ∈ ( bday ‘𝐵) ↔ (( bday ‘𝐴) +no ( bday ‘𝑌)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
28	24, 25, 26, 27	mp3an 1463	. . . . . 6 ⊢ (( bday ‘𝑌) ∈ ( bday ‘𝐵) ↔ (( bday ‘𝐴) +no ( bday ‘𝑌)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
29	23, 28	sylib 218	. . . . 5 ⊢ (𝜑 → (( bday ‘𝐴) +no ( bday ‘𝑌)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
30		elun1 4131	. . . . 5 ⊢ ((( bday ‘𝐴) +no ( bday ‘𝑌)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) → (( bday ‘𝐴) +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))))))
31	29, 30	syl 17	. . . 4 ⊢ (𝜑 → (( bday ‘𝐴) +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	21, 31	eqeltrid 2837	. . 3 ⊢ (𝜑 → ((( bday ‘𝐴) +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))))))
33	1, 2, 5, 7, 7, 7, 7, 32	mulsproplem1 28075	. 2 ⊢ (𝜑 → ((𝐴 ·s 𝑌) ∈ 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 → (𝐴 ·s 𝑌) ∈ No )

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048   ∪ cun 3896  ∅c0 4282   class class class wbr 5095  Oncon0 6314  ‘cfv 6489  (class class class)co 7355   +no cnadd 8589   No csur 27598   <s cslt 27599   bday cbday 27600   0_s c0s 27786   O cold 27804   -_s csubs 27982   ·_s cmuls 28065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-1o 8394  df-2o 8395  df-nadd 8590  df-no 27601  df-slt 27602  df-bday 27603  df-sslt 27741  df-scut 27743  df-0s 27788  df-made 27808  df-old 27809

This theorem is referenced by:  mulsproplem5  28079  mulsproplem6  28080  mulsproplem7  28081  mulsproplem8  28082  mulsproplem9  28083  mulsproplem14  28088