Theorem mulsproplem11 28569

Description: Lemma for surreal multiplication. Under the inductive hypothesis, demonstrate closure of surreal multiplication. (Contributed by Scott Fenton, 5-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))))))
mulsproplem9.1	⊢ (𝜑 → 𝐴 ∈ No)
mulsproplem9.2	⊢ (𝜑 → 𝐵 ∈ No)

Assertion

Ref	Expression
mulsproplem11	⊢ (𝜑 → (𝐴 ·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

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

Proof of Theorem mulsproplem11

Dummy variables 𝑔 ℎ 𝑖 𝑗 𝑝 𝑞 𝑟 𝑠 𝑡 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

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		mulsproplem9.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ No)
3		mulsproplem9.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ No)
4	1, 2, 3	mulsproplem10 28568	. 2 ⊢ (𝜑 → ((𝐴 ·s 𝐵) ∈ No ∧ ({𝑔 ∣ ∃𝑝 ∈ (L‘𝐴)∃𝑞 ∈ (L‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞))} ∪ {ℎ ∣ ∃𝑟 ∈ (R‘𝐴)∃𝑠 ∈ (R‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠))}) <<s {(𝐴 ·s 𝐵)} ∧ {(𝐴 ·s 𝐵)} <<s ({𝑖 ∣ ∃𝑡 ∈ (L‘𝐴)∃𝑢 ∈ (R‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑗 ∣ ∃𝑣 ∈ (R‘𝐴)∃𝑤 ∈ (L‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))})))
5	4	simp1d 1160	1 ⊢ (𝜑 → (𝐴 ·s 𝐵) ∈ No)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ∪ cun 4152  {csn 4588   class class class wbr 5108  ‘cfv 6566  (class class class)co 7451   +no cnadd 8712  Nocsur 28054   <s clts 28055  bdaycbday 28056   <<s cslts 28200  Lcleft 28268  Rcright 28269   +_s cadds 28402   −_s csubs 28463   ·_s cmuls 28549

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-nadd 8713  df-no 28057  df-lts 28058  df-bday 28059  df-les 28159  df-slts 28201  df-cuts 28203  df-0s 28250  df-made 28270  df-old 28271  df-left 28273  df-right 28274  df-norec 28381  df-norec2 28392  df-adds 28403  df-negs 28464  df-subs 28465  df-muls 28550

This theorem is used by:  mulsproplem13  28571  mulsproplem14  28572  mulsprop  28573