Theorem mulcut 28524

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 28523	. 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 264	. . 3 ⊢ ((𝐴 ·s 𝐵) ∈ No ↔ (𝐴 ·s 𝐵) ∈ No)
5		mulsval2lem 28502	. . . . 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 28502	. . . . 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 4165	. . . 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 5109	. . 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 28502	. . . . 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 28502	. . . . 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 4165	. . . 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 5110	. . 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 1173	. 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 237	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))})))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  {cab 2738  ∃wrex 3086   ∪ cun 4149  {csn 4584   class class class wbr 5102  ‘cfv 6539  (class class class)co 7421  Nocsur 28003   <<s cslts 28149  Lcleft 28217  Rcright 28218   +_s cadds 28351   −_s csubs 28412   ·_s cmuls 28498

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-1o 8480  df-2o 8481  df-nadd 8679  df-no 28006  df-lts 28007  df-bday 28008  df-les 28108  df-slts 28150  df-cuts 28152  df-0s 28199  df-made 28219  df-old 28220  df-left 28222  df-right 28223  df-norec 28330  df-norec2 28341  df-adds 28352  df-negs 28413  df-subs 28414  df-muls 28499

This theorem is used by:  mulcut2  28525