Theorem mulsunif 27534

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 27533	. 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 27495	. . . 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 27495	. . . 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 4158	. . 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 27495	. . . 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 27495	. . . 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 4158	. . 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 7406	. 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 2790	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 1541  {cab 2709  ∃wrex 3070   ∪ cun 3943   class class class wbr 5142  (class class class)co 7394   <<s csslt 27211   |s cscut 27213   +_s cadds 27372   -_s csubs 27424   ·_s cmuls 27491

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-ot 4632  df-uni 4903  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-se 5626  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7350  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7959  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-1o 8450  df-2o 8451  df-nadd 8650  df-no 27075  df-slt 27076  df-bday 27077  df-sle 27177  df-sslt 27212  df-scut 27214  df-0s 27254  df-made 27271  df-old 27272  df-left 27274  df-right 27275  df-norec 27351  df-norec2 27362  df-adds 27373  df-negs 27425  df-subs 27426  df-muls 27492

This theorem is referenced by:  addsdilem1  27535  mulsasslem1  27547  mulsasslem2  27548  precsexlem11  27592