Theorem mulsproplem4 28160

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 27915	. . . 4 ⊢ ( O ‘( bday ‘𝐴)) ⊆ No
3		mulsproplem4.1	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ( O ‘( bday ‘𝐴)))
4	2, 3	sselid 3993	. . 3 ⊢ (𝜑 → 𝑋 ∈ No )
5		oldssno 27915	. . . 4 ⊢ ( O ‘( bday ‘𝐵)) ⊆ No
6		mulsproplem4.2	. . . 4 ⊢ (𝜑 → 𝑌 ∈ ( O ‘( bday ‘𝐵)))
7	5, 6	sselid 3993	. . 3 ⊢ (𝜑 → 𝑌 ∈ No )
8		0sno 27886	. . . 4 ⊢ 0s ∈ No
9	8	a1i 11	. . 3 ⊢ (𝜑 → 0s ∈ No )
10		bday0s 27888	. . . . . . . . . . . 12 ⊢ ( bday ‘ 0s ) = ∅
11	10, 10	oveq12i 7443	. . . . . . . . . . 11 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = (∅ +no ∅)
12		0elon 6440	. . . . . . . . . . . 12 ⊢ ∅ ∈ On
13		naddrid 8720	. . . . . . . . . . . 12 ⊢ (∅ ∈ On → (∅ +no ∅) = ∅)
14	12, 13	ax-mp 5	. . . . . . . . . . 11 ⊢ (∅ +no ∅) = ∅
15	11, 14	eqtri 2763	. . . . . . . . . 10 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = ∅
16	15, 15	uneq12i 4176	. . . . . . . . 9 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = (∅ ∪ ∅)
17		un0 4400	. . . . . . . . 9 ⊢ (∅ ∪ ∅) = ∅
18	16, 17	eqtri 2763	. . . . . . . 8 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = ∅
19	18, 18	uneq12i 4176	. . . . . . 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 2763	. . . . . 6 ⊢ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))) = ∅
21	20	uneq2i 4175	. . . . 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 4400	. . . . 5 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ ∅) = (( bday ‘𝑋) +no ( bday ‘𝑌))
23	21, 22	eqtri 2763	. . . 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 27942	. . . . . . 7 ⊢ (𝑋 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑋) ∈ ( bday ‘𝐴))
25	3, 24	syl 17	. . . . . 6 ⊢ (𝜑 → ( bday ‘𝑋) ∈ ( bday ‘𝐴))
26		oldbdayim 27942	. . . . . . 7 ⊢ (𝑌 ∈ ( O ‘( bday ‘𝐵)) → ( bday ‘𝑌) ∈ ( bday ‘𝐵))
27	6, 26	syl 17	. . . . . 6 ⊢ (𝜑 → ( bday ‘𝑌) ∈ ( bday ‘𝐵))
28		bdayelon 27836	. . . . . . 7 ⊢ ( bday ‘𝐴) ∈ On
29		bdayelon 27836	. . . . . . 7 ⊢ ( bday ‘𝐵) ∈ On
30		naddel12 8737	. . . . . . 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 692	. . . . . 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 4192	. . . . 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 2843	. . 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 28157	. 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 494	1 ⊢ (𝜑 → (𝑋 ·s 𝑌) ∈ No )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059   ∪ cun 3961  ∅c0 4339   class class class wbr 5148  Oncon0 6386  ‘cfv 6563  (class class class)co 7431   +no cnadd 8702   No csur 27699   <s cslt 27700   bday cbday 27701   0_s c0s 27882   O cold 27897   -_s csubs 28067   ·_s cmuls 28147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-0s 27884  df-made 27901  df-old 27902

This theorem is referenced by:  mulsproplem5  28161  mulsproplem6  28162  mulsproplem7  28163  mulsproplem8  28164  mulsproplem9  28165