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Theorem mulscutlem 28172
Description: Lemma for mulscut 28173. 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 28171 . . . . . . . . 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 1117 . . . . . . 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 482 . . 3 ((𝐴 No 𝐵 No ) → 𝐴 No )
11 simpr 484 . . 3 ((𝐴 No 𝐵 No ) → 𝐵 No )
129, 10, 11mulsproplem10 28166 . 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 395  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  cun 3961  {csn 4631   class class class wbr 5148  cfv 6563  (class class class)co 7431   +no cnadd 8702   No csur 27699   <s cslt 27700   bday cbday 27701   <<s csslt 27840   0s c0s 27882   L cleft 27899   R cright 27900   +s cadds 28007   -s csubs 28067   ·s cmuls 28147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-muls 28148
This theorem is referenced by:  mulscut  28173
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