Theorem mulsval2 28045

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 28043	. 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 28044	. . . 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 28044	. . . 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 4111	. . 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 28044	. . . 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 28044	. . . 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 4111	. . 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 7353	. 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 2784	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))})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  ∃wrex 3056   ∪ cun 3895  ‘cfv 6476  (class class class)co 7341   No csur 27573   |s cscut 27717   L cleft 27781   R cright 27782   +_s cadds 27897   -_s csubs 27957   ·_s cmuls 28040

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-se 5565  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-1o 8380  df-2o 8381  df-no 27576  df-slt 27577  df-bday 27578  df-sslt 27716  df-scut 27718  df-made 27783  df-old 27784  df-left 27786  df-right 27787  df-norec2 27887  df-muls 28041

This theorem is referenced by:  addsdilem2  28086  mulsasslem1  28097  mulsasslem2  28098