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Theorem mulsproplem3 27933
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 𝑑𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
mulsproplem3.1 (𝜑𝐴 No )
mulsproplem3.2 (𝜑𝑌 ∈ ( 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
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 mulsproplem3.1 . . 3 (𝜑𝐴 No )
3 oldssno 27703 . . . 4 ( O ‘( bday 𝐵)) ⊆ No
4 mulsproplem3.2 . . . 4 (𝜑𝑌 ∈ ( O ‘( bday 𝐵)))
53, 4sselid 3980 . . 3 (𝜑𝑌 No )
6 0sno 27674 . . . 4 0s No
76a1i 11 . . 3 (𝜑 → 0s No )
8 bday0s 27676 . . . . . . . . . . . 12 ( bday ‘ 0s ) = ∅
98, 8oveq12i 7424 . . . . . . . . . . 11 (( bday ‘ 0s ) +no ( bday ‘ 0s )) = (∅ +no ∅)
10 0elon 6418 . . . . . . . . . . . 12 ∅ ∈ On
11 naddrid 8688 . . . . . . . . . . . 12 (∅ ∈ On → (∅ +no ∅) = ∅)
1210, 11ax-mp 5 . . . . . . . . . . 11 (∅ +no ∅) = ∅
139, 12eqtri 2759 . . . . . . . . . 10 (( bday ‘ 0s ) +no ( bday ‘ 0s )) = ∅
1413, 13uneq12i 4161 . . . . . . . . 9 ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = (∅ ∪ ∅)
15 un0 4390 . . . . . . . . 9 (∅ ∪ ∅) = ∅
1614, 15eqtri 2759 . . . . . . . 8 ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = ∅
1716, 16uneq12i 4161 . . . . . . 7 (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))) = (∅ ∪ ∅)
1817, 15eqtri 2759 . . . . . 6 (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))) = ∅
1918uneq2i 4160 . . . . 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 4390 . . . . 5 ((( bday 𝐴) +no ( bday 𝑌)) ∪ ∅) = (( bday 𝐴) +no ( bday 𝑌))
2119, 20eqtri 2759 . . . 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 27730 . . . . . . 7 (𝑌 ∈ ( O ‘( bday 𝐵)) → ( bday 𝑌) ∈ ( bday 𝐵))
234, 22syl 17 . . . . . 6 (𝜑 → ( bday 𝑌) ∈ ( bday 𝐵))
24 bdayelon 27624 . . . . . . 7 ( bday 𝑌) ∈ On
25 bdayelon 27624 . . . . . . 7 ( bday 𝐵) ∈ On
26 bdayelon 27624 . . . . . . 7 ( bday 𝐴) ∈ On
27 naddel2 8693 . . . . . . 7 ((( bday 𝑌) ∈ On ∧ ( bday 𝐵) ∈ On ∧ ( bday 𝐴) ∈ On) → (( bday 𝑌) ∈ ( bday 𝐵) ↔ (( bday 𝐴) +no ( bday 𝑌)) ∈ (( bday 𝐴) +no ( bday 𝐵))))
2824, 25, 26, 27mp3an 1460 . . . . . 6 (( bday 𝑌) ∈ ( bday 𝐵) ↔ (( bday 𝐴) +no ( bday 𝑌)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
2923, 28sylib 217 . . . . 5 (𝜑 → (( bday 𝐴) +no ( bday 𝑌)) ∈ (( bday 𝐴) +no ( bday 𝐵)))
30 elun1 4176 . . . . 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))))))
3129, 30syl 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))))))
3221, 31eqeltrid 2836 . . 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))))))
331, 2, 5, 7, 7, 7, 7, 32mulsproplem1 27931 . 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 )))))
3433simpld 494 1 (𝜑 → (𝐴 ·s 𝑌) ∈ No )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wcel 2105  wral 3060  cun 3946  c0 4322   class class class wbr 5148  Oncon0 6364  cfv 6543  (class class class)co 7412   +no cnadd 8670   No csur 27488   <s cslt 27489   bday cbday 27490   0s c0s 27670   O cold 27685   -s csubs 27848   ·s cmuls 27921
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-1o 8472  df-2o 8473  df-nadd 8671  df-no 27491  df-slt 27492  df-bday 27493  df-sslt 27629  df-scut 27631  df-0s 27672  df-made 27689  df-old 27690
This theorem is referenced by:  mulsproplem5  27935  mulsproplem6  27936  mulsproplem7  27937  mulsproplem8  27938  mulsproplem9  27939  mulsproplem14  27944
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