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Theorem mulsproplem2 28158
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 𝑑𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
mulsproplem2.1 (𝜑𝑋 ∈ ( O ‘( bday 𝐴)))
mulsproplem2.2 (𝜑𝐵 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
StepHypRef 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 𝑑𝑒 <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 mulsproplem2.1 . . . 4 (𝜑𝑋 ∈ ( O ‘( bday 𝐴)))
42, 3sselid 3993 . . 3 (𝜑𝑋 No )
5 mulsproplem2.2 . . 3 (𝜑𝐵 No )
6 0sno 27886 . . . 4 0s No
76a1i 11 . . 3 (𝜑 → 0s No )
8 bday0s 27888 . . . . . . . . . . . 12 ( bday ‘ 0s ) = ∅
98, 8oveq12i 7443 . . . . . . . . . . 11 (( bday ‘ 0s ) +no ( bday ‘ 0s )) = (∅ +no ∅)
10 0elon 6440 . . . . . . . . . . . 12 ∅ ∈ On
11 naddrid 8720 . . . . . . . . . . . 12 (∅ ∈ On → (∅ +no ∅) = ∅)
1210, 11ax-mp 5 . . . . . . . . . . 11 (∅ +no ∅) = ∅
139, 12eqtri 2763 . . . . . . . . . 10 (( bday ‘ 0s ) +no ( bday ‘ 0s )) = ∅
1413, 13uneq12i 4176 . . . . . . . . 9 ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = (∅ ∪ ∅)
15 un0 4400 . . . . . . . . 9 (∅ ∪ ∅) = ∅
1614, 15eqtri 2763 . . . . . . . 8 ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = ∅
1716, 16uneq12i 4176 . . . . . . 7 (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))) = (∅ ∪ ∅)
1817, 15eqtri 2763 . . . . . 6 (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))) = ∅
1918uneq2i 4175 . . . . 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 4400 . . . . 5 ((( bday 𝑋) +no ( bday 𝐵)) ∪ ∅) = (( bday 𝑋) +no ( bday 𝐵))
2119, 20eqtri 2763 . . . 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 27942 . . . . . . 7 (𝑋 ∈ ( O ‘( bday 𝐴)) → ( bday 𝑋) ∈ ( bday 𝐴))
233, 22syl 17 . . . . . 6 (𝜑 → ( bday 𝑋) ∈ ( bday 𝐴))
24 bdayelon 27836 . . . . . . 7 ( bday 𝑋) ∈ On
25 bdayelon 27836 . . . . . . 7 ( bday 𝐴) ∈ On
26 bdayelon 27836 . . . . . . 7 ( bday 𝐵) ∈ On
27 naddel1 8724 . . . . . . 7 ((( bday 𝑋) ∈ On ∧ ( bday 𝐴) ∈ On ∧ ( bday 𝐵) ∈ On) → (( bday 𝑋) ∈ ( bday 𝐴) ↔ (( bday 𝑋) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
2824, 25, 26, 27mp3an 1460 . . . . . 6 (( bday 𝑋) ∈ ( bday 𝐴) ↔ (( bday 𝑋) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
2923, 28sylib 218 . . . . 5 (𝜑 → (( bday 𝑋) +no ( bday 𝐵)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
30 elun1 4192 . . . . 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))))))
3129, 30syl 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))))))
3221, 31eqeltrid 2843 . . 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))))))
331, 4, 5, 7, 7, 7, 7, 32mulsproplem1 28157 . 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 )))))
3433simpld 494 1 (𝜑 → (𝑋 ·s 𝐵) ∈ No )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  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   0s 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  mulsproplem13  28169
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