Theorem mulcut 28128

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

Hypotheses

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

Assertion

Ref	Expression
mulcut	⊢ (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 mulcut

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		mulcut.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
2		mulcut.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ No )
3	1, 2	mulcutlem 28127	. 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 261	. . 3 ⊢ ((𝐴 ·s 𝐵) ∈ No ↔ (𝐴 ·s 𝐵) ∈ No )
5		mulsval2lem 28106	. . . . 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 28106	. . . . 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 4118	. . . 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 5105	. . 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 28106	. . . . 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 28106	. . . . 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 4118	. . . 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 5106	. . 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 234	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 1086   = wceq 1541   ∈ wcel 2113  {cab 2714  ∃wrex 3060   ∪ cun 3899  {csn 4580   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358   No csur 27607   <<s cslts 27753   L cleft 27821   R cright 27822   +_s cadds 27955   -_s csubs 28016   ·_s cmuls 28102

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-ot 4589  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-1o 8397  df-2o 8398  df-nadd 8594  df-no 27610  df-lts 27611  df-bday 27612  df-les 27713  df-slts 27754  df-cuts 27756  df-0s 27803  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-norec 27934  df-norec2 27945  df-adds 27956  df-negs 28017  df-subs 28018  df-muls 28103

This theorem is referenced by:  mulcut2  28129