Theorem mulsval2 28554

Description: The value of surreal multiplication, expressed with fewer distinct variable conditions. (Contributed by Scott Fenton, 8-Mar-2025.)

Assertion

Ref	Expression
mulsval2	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ (L‘𝐴)∃𝑞 ∈ (L‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ (R‘𝐴)∃𝑠 ∈ (R‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ (L‘𝐴)∃𝑢 ∈ (R‘𝐵)𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ (R‘𝐴)∃𝑤 ∈ (L‘𝐵)𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))})))

Distinct variable groups:   𝐴,𝑎,𝑝,𝑞   𝐴,𝑏,𝑟,𝑠   𝐴,𝑐,𝑡,𝑢   𝐴,𝑑,𝑣,𝑤   𝐵,𝑎,𝑝,𝑞   𝐵,𝑏,𝑟,𝑠   𝐵,𝑐,𝑡,𝑢   𝐵,𝑑,𝑣,𝑤

Proof of Theorem mulsval2

Dummy variables 𝑒 𝑓 𝑔 ℎ 𝑖 𝑘 𝑙 𝑚 𝑛 𝑜 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulsval 28552	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 ·s 𝐵) = (({𝑒 ∣ ∃𝑓 ∈ (L‘𝐴)∃𝑔 ∈ (L‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) −s (𝑓 ·s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ (R‘𝐴)∃𝑘 ∈ (R‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑘)) −s (𝑖 ·s 𝑘))}) |s ({𝑙 ∣ ∃𝑚 ∈ (L‘𝐴)∃𝑛 ∈ (R‘𝐵)𝑙 = (((𝑚 ·s 𝐵) +s (𝐴 ·s 𝑛)) −s (𝑚 ·s 𝑛))} ∪ {𝑜 ∣ ∃𝑥 ∈ (R‘𝐴)∃𝑦 ∈ (L‘𝐵)𝑜 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) −s (𝑥 ·s 𝑦))})))
2		mulsval2lem 28553	. . . 4 ⊢ {𝑎 ∣ ∃𝑝 ∈ (L‘𝐴)∃𝑞 ∈ (L‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞))} = {𝑒 ∣ ∃𝑓 ∈ (L‘𝐴)∃𝑔 ∈ (L‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) −s (𝑓 ·s 𝑔))}
3		mulsval2lem 28553	. . . 4 ⊢ {𝑏 ∣ ∃𝑟 ∈ (R‘𝐴)∃𝑠 ∈ (R‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠))} = {ℎ ∣ ∃𝑖 ∈ (R‘𝐴)∃𝑘 ∈ (R‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑘)) −s (𝑖 ·s 𝑘))}
4	2, 3	uneq12i 4168	. . 3 ⊢ ({𝑎 ∣ ∃𝑝 ∈ (L‘𝐴)∃𝑞 ∈ (L‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ (R‘𝐴)∃𝑠 ∈ (R‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠))}) = ({𝑒 ∣ ∃𝑓 ∈ (L‘𝐴)∃𝑔 ∈ (L‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) −s (𝑓 ·s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ (R‘𝐴)∃𝑘 ∈ (R‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑘)) −s (𝑖 ·s 𝑘))})
5		mulsval2lem 28553	. . . 4 ⊢ {𝑐 ∣ ∃𝑡 ∈ (L‘𝐴)∃𝑢 ∈ (R‘𝐵)𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} = {𝑙 ∣ ∃𝑚 ∈ (L‘𝐴)∃𝑛 ∈ (R‘𝐵)𝑙 = (((𝑚 ·s 𝐵) +s (𝐴 ·s 𝑛)) −s (𝑚 ·s 𝑛))}
6		mulsval2lem 28553	. . . 4 ⊢ {𝑑 ∣ ∃𝑣 ∈ (R‘𝐴)∃𝑤 ∈ (L‘𝐵)𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))} = {𝑜 ∣ ∃𝑥 ∈ (R‘𝐴)∃𝑦 ∈ (L‘𝐵)𝑜 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) −s (𝑥 ·s 𝑦))}
7	5, 6	uneq12i 4168	. . 3 ⊢ ({𝑐 ∣ ∃𝑡 ∈ (L‘𝐴)∃𝑢 ∈ (R‘𝐵)𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ (R‘𝐴)∃𝑤 ∈ (L‘𝐵)𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))}) = ({𝑙 ∣ ∃𝑚 ∈ (L‘𝐴)∃𝑛 ∈ (R‘𝐵)𝑙 = (((𝑚 ·s 𝐵) +s (𝐴 ·s 𝑛)) −s (𝑚 ·s 𝑛))} ∪ {𝑜 ∣ ∃𝑥 ∈ (R‘𝐴)∃𝑦 ∈ (L‘𝐵)𝑜 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) −s (𝑥 ·s 𝑦))})
8	4, 7	oveq12i 7463	. 2 ⊢ (({𝑎 ∣ ∃𝑝 ∈ (L‘𝐴)∃𝑞 ∈ (L‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ (R‘𝐴)∃𝑠 ∈ (R‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ (L‘𝐴)∃𝑢 ∈ (R‘𝐵)𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ (R‘𝐴)∃𝑤 ∈ (L‘𝐵)𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))})) = (({𝑒 ∣ ∃𝑓 ∈ (L‘𝐴)∃𝑔 ∈ (L‘𝐵)𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) −s (𝑓 ·s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ (R‘𝐴)∃𝑘 ∈ (R‘𝐵)ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑘)) −s (𝑖 ·s 𝑘))}) |s ({𝑙 ∣ ∃𝑚 ∈ (L‘𝐴)∃𝑛 ∈ (R‘𝐵)𝑙 = (((𝑚 ·s 𝐵) +s (𝐴 ·s 𝑛)) −s (𝑚 ·s 𝑛))} ∪ {𝑜 ∣ ∃𝑥 ∈ (R‘𝐴)∃𝑦 ∈ (L‘𝐵)𝑜 = (((𝑥 ·s 𝐵) +s (𝐴 ·s 𝑦)) −s (𝑥 ·s 𝑦))}))
9	1, 8	eqtr4di 2816	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ (L‘𝐴)∃𝑞 ∈ (L‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) −s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ (R‘𝐴)∃𝑠 ∈ (R‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) −s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ (L‘𝐴)∃𝑢 ∈ (R‘𝐵)𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) −s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ (R‘𝐴)∃𝑤 ∈ (L‘𝐵)𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) −s (𝑣 ·s 𝑤))})))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090   ∪ cun 4152  ‘cfv 6566  (class class class)co 7451  Nocsur 28054   |s ccuts 28202  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-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-no 28057  df-lts 28058  df-bday 28059  df-slts 28201  df-cuts 28203  df-made 28270  df-old 28271  df-left 28273  df-right 28274  df-norec2 28392  df-muls 28550

This theorem is used by:  addsdilem2  28595  mulsasslem1  28606  mulsasslem2  28607