Theorem mulsunif 28142

Description: Surreal multiplication has the uniformity property. That is, any cuts that define 𝐴 and 𝐵 can be used in the definition of (𝐴 ·_s 𝐵). Theorem 3.5 of [Gonshor] p. 18. (Contributed by Scott Fenton, 7-Mar-2025.)

Hypotheses

Ref	Expression
mulsunif.1	⊢ (𝜑 → 𝐿 <<s 𝑅)
mulsunif.2	⊢ (𝜑 → 𝑀 <<s 𝑆)
mulsunif.3	⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))
mulsunif.4	⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))

Assertion

Ref	Expression
mulsunif	⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 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   𝐿,𝑎,𝑝   𝐿,𝑐,𝑡   𝑀,𝑎,𝑝,𝑞   𝑀,𝑑,𝑣,𝑤   𝑅,𝑏   𝑅,𝑑   𝑅,𝑟   𝑣,𝑅   𝑆,𝑏   𝑆,𝑐   𝑆,𝑟,𝑠   𝑡,𝑆,𝑢

Allowed substitution hints:   𝜑(𝑤,𝑣,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝,𝑎,𝑏,𝑐,𝑑)   𝑅(𝑤,𝑢,𝑡,𝑠,𝑞,𝑝,𝑎,𝑐)   𝑆(𝑤,𝑣,𝑞,𝑝,𝑎,𝑑)   𝐿(𝑤,𝑣,𝑢,𝑠,𝑟,𝑞,𝑏,𝑑)   𝑀(𝑢,𝑡,𝑠,𝑟,𝑏,𝑐)

Proof of Theorem mulsunif

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

Step	Hyp	Ref	Expression
1		mulsunif.1	. . 3 ⊢ (𝜑 → 𝐿 <<s 𝑅)
2		mulsunif.2	. . 3 ⊢ (𝜑 → 𝑀 <<s 𝑆)
3		mulsunif.3	. . 3 ⊢ (𝜑 → 𝐴 = (𝐿 |s 𝑅))
4		mulsunif.4	. . 3 ⊢ (𝜑 → 𝐵 = (𝑀 |s 𝑆))
5	1, 2, 3, 4	mulsuniflem 28141	. 2 ⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑒 ∣ ∃𝑓 ∈ 𝐿 ∃𝑔 ∈ 𝑀 𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ 𝑅 ∃𝑗 ∈ 𝑆 ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))}) |s ({𝑘 ∣ ∃𝑙 ∈ 𝐿 ∃𝑚 ∈ 𝑆 𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))} ∪ {𝑛 ∣ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 𝑛 = (((𝑜 ·s 𝐵) +s (𝐴 ·s 𝑥)) -s (𝑜 ·s 𝑥))})))
6		mulsval2lem 28102	. . . 4 ⊢ {𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} = {𝑒 ∣ ∃𝑓 ∈ 𝐿 ∃𝑔 ∈ 𝑀 𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))}
7		mulsval2lem 28102	. . . 4 ⊢ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} = {ℎ ∣ ∃𝑖 ∈ 𝑅 ∃𝑗 ∈ 𝑆 ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))}
8	6, 7	uneq12i 4106	. . 3 ⊢ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) = ({𝑒 ∣ ∃𝑓 ∈ 𝐿 ∃𝑔 ∈ 𝑀 𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ 𝑅 ∃𝑗 ∈ 𝑆 ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))})
9		mulsval2lem 28102	. . . 4 ⊢ {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} = {𝑘 ∣ ∃𝑙 ∈ 𝐿 ∃𝑚 ∈ 𝑆 𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))}
10		mulsval2lem 28102	. . . 4 ⊢ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} = {𝑛 ∣ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 𝑛 = (((𝑜 ·s 𝐵) +s (𝐴 ·s 𝑥)) -s (𝑜 ·s 𝑥))}
11	9, 10	uneq12i 4106	. . 3 ⊢ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) = ({𝑘 ∣ ∃𝑙 ∈ 𝐿 ∃𝑚 ∈ 𝑆 𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))} ∪ {𝑛 ∣ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 𝑛 = (((𝑜 ·s 𝐵) +s (𝐴 ·s 𝑥)) -s (𝑜 ·s 𝑥))})
12	8, 11	oveq12i 7379	. 2 ⊢ (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})) = (({𝑒 ∣ ∃𝑓 ∈ 𝐿 ∃𝑔 ∈ 𝑀 𝑒 = (((𝑓 ·s 𝐵) +s (𝐴 ·s 𝑔)) -s (𝑓 ·s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ 𝑅 ∃𝑗 ∈ 𝑆 ℎ = (((𝑖 ·s 𝐵) +s (𝐴 ·s 𝑗)) -s (𝑖 ·s 𝑗))}) |s ({𝑘 ∣ ∃𝑙 ∈ 𝐿 ∃𝑚 ∈ 𝑆 𝑘 = (((𝑙 ·s 𝐵) +s (𝐴 ·s 𝑚)) -s (𝑙 ·s 𝑚))} ∪ {𝑛 ∣ ∃𝑜 ∈ 𝑅 ∃𝑥 ∈ 𝑀 𝑛 = (((𝑜 ·s 𝐵) +s (𝐴 ·s 𝑥)) -s (𝑜 ·s 𝑥))}))
13	5, 12	eqtr4di 2789	1 ⊢ (𝜑 → (𝐴 ·s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) |s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})))

Syntax hints:   → wi 4   = wceq 1542  {cab 2714  ∃wrex 3061   ∪ cun 3887   class class class wbr 5085  (class class class)co 7367   <<s cslts 27749   |s ccuts 27751   +_s cadds 27951   -_s csubs 28012   ·_s cmuls 28098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-2o 8406  df-nadd 8602  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-0s 27799  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-norec 27930  df-norec2 27941  df-adds 27952  df-negs 28013  df-subs 28014  df-muls 28099

This theorem is referenced by:  addsdilem1  28143  mulsasslem1  28155  mulsasslem2  28156  mulsunif2lem  28161  precsexlem11  28209  onmulscl  28270