Theorem mulsval2 28504

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 28502	. 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 28503	. . . 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 28503	. . . 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 4164	. . 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 28503	. . . 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 28503	. . . 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 4164	. . 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 7434	. 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 2813	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 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086   ∪ cun 4148  ‘cfv 6541  (class class class)co 7422  Nocsur 28004   |s ccuts 28152  Lcleft 28218  Rcright 28219   +_s cadds 28352   −_s csubs 28413   ·_s cmuls 28499

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-1st 8003  df-2nd 8004  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-1o 8481  df-2o 8482  df-no 28007  df-lts 28008  df-bday 28009  df-slts 28151  df-cuts 28153  df-made 28220  df-old 28221  df-left 28223  df-right 28224  df-norec2 28342  df-muls 28500

This theorem is used by:  addsdilem2  28545  mulsasslem1  28556  mulsasslem2  28557