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Theorem mulscutlem 28158
Description: Lemma for mulscut 28159. 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 28157 . . . . . . . . 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 3201 . . . . . 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 3201 . . . . 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 3198 . . . 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 28152 . 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 1539  wcel 2107  {cab 2713  wral 3060  wrex 3069  cun 3948  {csn 4625   class class class wbr 5142  cfv 6560  (class class class)co 7432   +no cnadd 8704   No csur 27685   <s cslt 27686   bday cbday 27687   <<s csslt 27826   0s c0s 27868   L cleft 27885   R cright 27886   +s cadds 27993   -s csubs 28053   ·s cmuls 28133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-ot 4634  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-1o 8507  df-2o 8508  df-nadd 8705  df-no 27688  df-slt 27689  df-bday 27690  df-sle 27791  df-sslt 27827  df-scut 27829  df-0s 27870  df-made 27887  df-old 27888  df-left 27890  df-right 27891  df-norec 27972  df-norec2 27983  df-adds 27994  df-negs 28054  df-subs 28055  df-muls 28134
This theorem is referenced by:  mulscut  28159
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