Theorem mulscut 27517

Description: Show the cut properties of surreal multiplication. (Contributed by Scott Fenton, 8-Mar-2025.)

Hypotheses

Ref	Expression
mulscut.1	⊢ (𝜑 → 𝐴 ∈ No )
mulscut.2	⊢ (𝜑 → 𝐵 ∈ No )

Assertion

Ref	Expression
mulscut	⊢ (𝜑 → ((𝐴 ·s 𝐵) ∈ No ∧ ({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 {(𝐴 ·s 𝐵)} ∧ {(𝐴 ·s 𝐵)} <<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

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

Proof of Theorem mulscut

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

Step	Hyp	Ref	Expression
1		mulscut.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
2		mulscut.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ No )
3	1, 2	mulscutlem 27516	. 2 ⊢ (𝜑 → ((𝐴 ·s 𝐵) ∈ No ∧ ({𝑓 ∣ ∃𝑔 ∈ ( L ‘𝐴)∃ℎ ∈ ( L ‘𝐵)𝑓 = (((𝑔 ·s 𝐵) +s (𝐴 ·s ℎ)) -s (𝑔 ·s ℎ))} ∪ {𝑖 ∣ ∃𝑗 ∈ ( R ‘𝐴)∃𝑘 ∈ ( R ‘𝐵)𝑖 = (((𝑗 ·s 𝐵) +s (𝐴 ·s 𝑘)) -s (𝑗 ·s 𝑘))}) <<s {(𝐴 ·s 𝐵)} ∧ {(𝐴 ·s 𝐵)} <<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))})))
4		biid 260	. . 3 ⊢ ((𝐴 ·s 𝐵) ∈ No ↔ (𝐴 ·s 𝐵) ∈ No )
5		mulsval2lem 27495	. . . . 5 ⊢ {𝑎 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑎 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} = {𝑓 ∣ ∃𝑔 ∈ ( L ‘𝐴)∃ℎ ∈ ( L ‘𝐵)𝑓 = (((𝑔 ·s 𝐵) +s (𝐴 ·s ℎ)) -s (𝑔 ·s ℎ))}
6		mulsval2lem 27495	. . . . 5 ⊢ {𝑏 ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑏 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} = {𝑖 ∣ ∃𝑗 ∈ ( R ‘𝐴)∃𝑘 ∈ ( R ‘𝐵)𝑖 = (((𝑗 ·s 𝐵) +s (𝐴 ·s 𝑘)) -s (𝑗 ·s 𝑘))}
7	5, 6	uneq12i 4158	. . . 4 ⊢ ({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))}) = ({𝑓 ∣ ∃𝑔 ∈ ( L ‘𝐴)∃ℎ ∈ ( L ‘𝐵)𝑓 = (((𝑔 ·s 𝐵) +s (𝐴 ·s ℎ)) -s (𝑔 ·s ℎ))} ∪ {𝑖 ∣ ∃𝑗 ∈ ( R ‘𝐴)∃𝑘 ∈ ( R ‘𝐵)𝑖 = (((𝑗 ·s 𝐵) +s (𝐴 ·s 𝑘)) -s (𝑗 ·s 𝑘))})
8	7	breq1i 5149	. . 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))}) <<s {(𝐴 ·s 𝐵)} ↔ ({𝑓 ∣ ∃𝑔 ∈ ( L ‘𝐴)∃ℎ ∈ ( L ‘𝐵)𝑓 = (((𝑔 ·s 𝐵) +s (𝐴 ·s ℎ)) -s (𝑔 ·s ℎ))} ∪ {𝑖 ∣ ∃𝑗 ∈ ( R ‘𝐴)∃𝑘 ∈ ( R ‘𝐵)𝑖 = (((𝑗 ·s 𝐵) +s (𝐴 ·s 𝑘)) -s (𝑗 ·s 𝑘))}) <<s {(𝐴 ·s 𝐵)})
9		mulsval2lem 27495	. . . . 5 ⊢ {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))}
10		mulsval2lem 27495	. . . . 5 ⊢ {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))}
11	9, 10	uneq12i 4158	. . . 4 ⊢ ({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))})
12	11	breq2i 5150	. . 3 ⊢ ({(𝐴 ·s 𝐵)} <<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))}) ↔ {(𝐴 ·s 𝐵)} <<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))}))
13	4, 8, 12	3anbi123i 1155	. 2 ⊢ (((𝐴 ·s 𝐵) ∈ No ∧ ({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 {(𝐴 ·s 𝐵)} ∧ {(𝐴 ·s 𝐵)} <<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))})) ↔ ((𝐴 ·s 𝐵) ∈ No ∧ ({𝑓 ∣ ∃𝑔 ∈ ( L ‘𝐴)∃ℎ ∈ ( L ‘𝐵)𝑓 = (((𝑔 ·s 𝐵) +s (𝐴 ·s ℎ)) -s (𝑔 ·s ℎ))} ∪ {𝑖 ∣ ∃𝑗 ∈ ( R ‘𝐴)∃𝑘 ∈ ( R ‘𝐵)𝑖 = (((𝑗 ·s 𝐵) +s (𝐴 ·s 𝑘)) -s (𝑗 ·s 𝑘))}) <<s {(𝐴 ·s 𝐵)} ∧ {(𝐴 ·s 𝐵)} <<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))})))
14	3, 13	sylibr 233	1 ⊢ (𝜑 → ((𝐴 ·s 𝐵) ∈ No ∧ ({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 {(𝐴 ·s 𝐵)} ∧ {(𝐴 ·s 𝐵)} <<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   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2709  ∃wrex 3070   ∪ cun 3943  {csn 4623   class class class wbr 5142  ‘cfv 6533  (class class class)co 7394   No csur 27072   <<s csslt 27211   L cleft 27269   R cright 27270   +_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:  mulscut2  27518