Theorem mulsunif 28111

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 28110	. 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 28071	. . . 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 28071	. . . 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 4146	. . 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 28071	. . . 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 28071	. . . 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 4146	. . 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 7424	. 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 2787	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 1539  {cab 2712  ∃wrex 3059   ∪ cun 3929   class class class wbr 5123  (class class class)co 7412   <<s csslt 27760   |s cscut 27762   +_s cadds 27927   -_s csubs 27987   ·_s cmuls 28067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7736

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-tp 4611  df-op 4613  df-ot 4615  df-uni 4888  df-int 4927  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-se 5618  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6493  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-1o 8487  df-2o 8488  df-nadd 8685  df-no 27622  df-slt 27623  df-bday 27624  df-sle 27725  df-sslt 27761  df-scut 27763  df-0s 27804  df-made 27821  df-old 27822  df-left 27824  df-right 27825  df-norec 27906  df-norec2 27917  df-adds 27928  df-negs 27988  df-subs 27989  df-muls 28068

This theorem is referenced by:  addsdilem1  28112  mulsasslem1  28124  mulsasslem2  28125  mulsunif2lem  28130  precsexlem11  28176  onmulscl  28222