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Theorem mulscutlem 27516
Description: Lemma for mulscut 27517. State the theorem with extra DV conditions. (Contributed by Scott Fenton, 7-Mar-2025.)
Hypotheses
Ref Expression
mulscutlem.1 (𝜑𝐴 No )
mulscutlem.2 (𝜑𝐵 No )
Assertion
Ref Expression
mulscutlem (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,𝑏,𝑐,𝑑,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑣,𝑤   𝐵,𝑎,𝑏,𝑐,𝑑,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑣,𝑤
Allowed substitution hints:   𝜑(𝑤,𝑣,𝑢,𝑡,𝑠,𝑟,𝑞,𝑝,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem mulscutlem
Dummy variables 𝑒 𝑓 𝑔 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulscutlem.1 . 2 (𝜑𝐴 No )
2 mulscutlem.2 . 2 (𝜑𝐵 No )
3 mulsprop 27515 . . . . . . . . 9 (((𝑒 No 𝑓 No ) ∧ (𝑔 No No ) ∧ (𝑖 No 𝑗 No )) → ((𝑒 ·s 𝑓) ∈ No ∧ ((𝑔 <s 𝑖 <s 𝑗) → ((𝑔 ·s 𝑗) -s (𝑔 ·s 𝑖)) <s (( ·s 𝑗) -s ( ·s 𝑖)))))
43a1d 25 . . . . . . . 8 (((𝑒 No 𝑓 No ) ∧ (𝑔 No No ) ∧ (𝑖 No 𝑗 No )) → (((( bday 𝑒) +no ( bday 𝑓)) ∪ (((( bday 𝑔) +no ( bday 𝑖)) ∪ (( bday ) +no ( bday 𝑗))) ∪ ((( bday 𝑔) +no ( bday 𝑗)) ∪ (( bday ) +no ( bday 𝑖))))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((𝑒 ·s 𝑓) ∈ No ∧ ((𝑔 <s 𝑖 <s 𝑗) → ((𝑔 ·s 𝑗) -s (𝑔 ·s 𝑖)) <s (( ·s 𝑗) -s ( ·s 𝑖))))))
543expa 1118 . . . . . . 7 ((((𝑒 No 𝑓 No ) ∧ (𝑔 No No )) ∧ (𝑖 No 𝑗 No )) → (((( bday 𝑒) +no ( bday 𝑓)) ∪ (((( bday 𝑔) +no ( bday 𝑖)) ∪ (( bday ) +no ( bday 𝑗))) ∪ ((( bday 𝑔) +no ( bday 𝑗)) ∪ (( bday ) +no ( bday 𝑖))))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((𝑒 ·s 𝑓) ∈ No ∧ ((𝑔 <s 𝑖 <s 𝑗) → ((𝑔 ·s 𝑗) -s (𝑔 ·s 𝑖)) <s (( ·s 𝑗) -s ( ·s 𝑖))))))
65ralrimivva 3200 . . . . . 6 (((𝑒 No 𝑓 No ) ∧ (𝑔 No No )) → ∀𝑖 No 𝑗 No (((( bday 𝑒) +no ( bday 𝑓)) ∪ (((( bday 𝑔) +no ( bday 𝑖)) ∪ (( bday ) +no ( bday 𝑗))) ∪ ((( bday 𝑔) +no ( bday 𝑗)) ∪ (( bday ) +no ( bday 𝑖))))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((𝑒 ·s 𝑓) ∈ No ∧ ((𝑔 <s 𝑖 <s 𝑗) → ((𝑔 ·s 𝑗) -s (𝑔 ·s 𝑖)) <s (( ·s 𝑗) -s ( ·s 𝑖))))))
76ralrimivva 3200 . . . . 5 ((𝑒 No 𝑓 No ) → ∀𝑔 No No 𝑖 No 𝑗 No (((( bday 𝑒) +no ( bday 𝑓)) ∪ (((( bday 𝑔) +no ( bday 𝑖)) ∪ (( bday ) +no ( bday 𝑗))) ∪ ((( bday 𝑔) +no ( bday 𝑗)) ∪ (( bday ) +no ( bday 𝑖))))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((𝑒 ·s 𝑓) ∈ No ∧ ((𝑔 <s 𝑖 <s 𝑗) → ((𝑔 ·s 𝑗) -s (𝑔 ·s 𝑖)) <s (( ·s 𝑗) -s ( ·s 𝑖))))))
87rgen2 3197 . . . 4 𝑒 No 𝑓 No 𝑔 No No 𝑖 No 𝑗 No (((( bday 𝑒) +no ( bday 𝑓)) ∪ (((( bday 𝑔) +no ( bday 𝑖)) ∪ (( bday ) +no ( bday 𝑗))) ∪ ((( bday 𝑔) +no ( bday 𝑗)) ∪ (( bday ) +no ( bday 𝑖))))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((𝑒 ·s 𝑓) ∈ No ∧ ((𝑔 <s 𝑖 <s 𝑗) → ((𝑔 ·s 𝑗) -s (𝑔 ·s 𝑖)) <s (( ·s 𝑗) -s ( ·s 𝑖)))))
98a1i 11 . . 3 ((𝐴 No 𝐵 No ) → ∀𝑒 No 𝑓 No 𝑔 No No 𝑖 No 𝑗 No (((( bday 𝑒) +no ( bday 𝑓)) ∪ (((( bday 𝑔) +no ( bday 𝑖)) ∪ (( bday ) +no ( bday 𝑗))) ∪ ((( bday 𝑔) +no ( bday 𝑗)) ∪ (( bday ) +no ( bday 𝑖))))) ∈ ((( bday 𝐴) +no ( bday 𝐵)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((𝑒 ·s 𝑓) ∈ No ∧ ((𝑔 <s 𝑖 <s 𝑗) → ((𝑔 ·s 𝑗) -s (𝑔 ·s 𝑖)) <s (( ·s 𝑗) -s ( ·s 𝑖))))))
10 simpl 483 . . 3 ((𝐴 No 𝐵 No ) → 𝐴 No )
11 simpr 485 . . 3 ((𝐴 No 𝐵 No ) → 𝐵 No )
129, 10, 11mulsproplem10 27510 . 2 ((𝐴 No 𝐵 No ) → ((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))})))
131, 2, 12syl2anc 584 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))})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  {cab 2709  wral 3061  wrex 3070  cun 3943  {csn 4623   class class class wbr 5142  cfv 6533  (class class class)co 7394   +no cnadd 8649   No csur 27072   <s cslt 27073   bday cbday 27074   <<s csslt 27211   0s c0s 27252   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:  mulscut  27517
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