Theorem mulscutlem 27500

Description: Lemma for mulscut 27501. State the theorem with extra DV conditions. (Contributed by Scott Fenton, 7-Mar-2025.)

Hypotheses

Ref	Expression
mulscutlem.1	⊢ (𝜑 → 𝐴 ∈ No )
mulscutlem.2	⊢ (𝜑 → 𝐵 ∈ 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 ℎ 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulscutlem.1	. 2 ⊢ (𝜑 → 𝐴 ∈ No )
2		mulscutlem.2	. 2 ⊢ (𝜑 → 𝐵 ∈ No )
3		mulsprop 27499	. . . . . . . . 9 ⊢ (((𝑒 ∈ No ∧ 𝑓 ∈ No ) ∧ (𝑔 ∈ No ∧ ℎ ∈ No ) ∧ (𝑖 ∈ No ∧ 𝑗 ∈ No )) → ((𝑒 ·s 𝑓) ∈ No ∧ ((𝑔 <s ℎ ∧ 𝑖 <s 𝑗) → ((𝑔 ·s 𝑗) -s (𝑔 ·s 𝑖)) <s ((ℎ ·s 𝑗) -s (ℎ ·s 𝑖)))))
4	3	a1d 25	. . . . . . . 8 ⊢ (((𝑒 ∈ No ∧ 𝑓 ∈ No ) ∧ (𝑔 ∈ No ∧ ℎ ∈ No ) ∧ (𝑖 ∈ No ∧ 𝑗 ∈ No )) → (((( bday ‘𝑒) +no ( bday ‘𝑓)) ∪ (((( bday ‘𝑔) +no ( bday ‘𝑖)) ∪ (( bday ‘ℎ) +no ( bday ‘𝑗))) ∪ ((( bday ‘𝑔) +no ( bday ‘𝑗)) ∪ (( bday ‘ℎ) +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 ℎ ∧ 𝑖 <s 𝑗) → ((𝑔 ·s 𝑗) -s (𝑔 ·s 𝑖)) <s ((ℎ ·s 𝑗) -s (ℎ ·s 𝑖))))))
5	4	3expa 1118	. . . . . . 7 ⊢ ((((𝑒 ∈ No ∧ 𝑓 ∈ No ) ∧ (𝑔 ∈ No ∧ ℎ ∈ No )) ∧ (𝑖 ∈ No ∧ 𝑗 ∈ No )) → (((( bday ‘𝑒) +no ( bday ‘𝑓)) ∪ (((( bday ‘𝑔) +no ( bday ‘𝑖)) ∪ (( bday ‘ℎ) +no ( bday ‘𝑗))) ∪ ((( bday ‘𝑔) +no ( bday ‘𝑗)) ∪ (( bday ‘ℎ) +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 ℎ ∧ 𝑖 <s 𝑗) → ((𝑔 ·s 𝑗) -s (𝑔 ·s 𝑖)) <s ((ℎ ·s 𝑗) -s (ℎ ·s 𝑖))))))
6	5	ralrimivva 3199	. . . . . 6 ⊢ (((𝑒 ∈ No ∧ 𝑓 ∈ No ) ∧ (𝑔 ∈ No ∧ ℎ ∈ No )) → ∀𝑖 ∈ No ∀𝑗 ∈ No (((( bday ‘𝑒) +no ( bday ‘𝑓)) ∪ (((( bday ‘𝑔) +no ( bday ‘𝑖)) ∪ (( bday ‘ℎ) +no ( bday ‘𝑗))) ∪ ((( bday ‘𝑔) +no ( bday ‘𝑗)) ∪ (( bday ‘ℎ) +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 ℎ ∧ 𝑖 <s 𝑗) → ((𝑔 ·s 𝑗) -s (𝑔 ·s 𝑖)) <s ((ℎ ·s 𝑗) -s (ℎ ·s 𝑖))))))
7	6	ralrimivva 3199	. . . . 5 ⊢ ((𝑒 ∈ No ∧ 𝑓 ∈ No ) → ∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No (((( bday ‘𝑒) +no ( bday ‘𝑓)) ∪ (((( bday ‘𝑔) +no ( bday ‘𝑖)) ∪ (( bday ‘ℎ) +no ( bday ‘𝑗))) ∪ ((( bday ‘𝑔) +no ( bday ‘𝑗)) ∪ (( bday ‘ℎ) +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 ℎ ∧ 𝑖 <s 𝑗) → ((𝑔 ·s 𝑗) -s (𝑔 ·s 𝑖)) <s ((ℎ ·s 𝑗) -s (ℎ ·s 𝑖))))))
8	7	rgen2 3196	. . . 4 ⊢ ∀𝑒 ∈ No ∀𝑓 ∈ No ∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No (((( bday ‘𝑒) +no ( bday ‘𝑓)) ∪ (((( bday ‘𝑔) +no ( bday ‘𝑖)) ∪ (( bday ‘ℎ) +no ( bday ‘𝑗))) ∪ ((( bday ‘𝑔) +no ( bday ‘𝑗)) ∪ (( bday ‘ℎ) +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 ℎ ∧ 𝑖 <s 𝑗) → ((𝑔 ·s 𝑗) -s (𝑔 ·s 𝑖)) <s ((ℎ ·s 𝑗) -s (ℎ ·s 𝑖)))))
9	8	a1i 11	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ∀𝑒 ∈ No ∀𝑓 ∈ No ∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No (((( bday ‘𝑒) +no ( bday ‘𝑓)) ∪ (((( bday ‘𝑔) +no ( bday ‘𝑖)) ∪ (( bday ‘ℎ) +no ( bday ‘𝑗))) ∪ ((( bday ‘𝑔) +no ( bday ‘𝑗)) ∪ (( bday ‘ℎ) +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 ℎ ∧ 𝑖 <s 𝑗) → ((𝑔 ·s 𝑗) -s (𝑔 ·s 𝑖)) <s ((ℎ ·s 𝑗) -s (ℎ ·s 𝑖))))))
10		simpl 483	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐴 ∈ No )
11		simpr 485	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ∈ No )
12	9, 10, 11	mulsproplem10 27494	. 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))})))
13	1, 2, 12	syl2anc 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))})))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2708  ∀wral 3060  ∃wrex 3069   ∪ cun 3942  {csn 4622   class class class wbr 5141  ‘cfv 6532  (class class class)co 7393   +no cnadd 8647   No csur 27070   <s cslt 27071   bday cbday 27072   <<s csslt 27208   0_s c0s 27249   L cleft 27263   R cright 27264   +_s cadds 27359   -_s csubs 27411   ·_s cmuls 27476

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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708

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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-tp 4627  df-op 4629  df-ot 4631  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-1o 8448  df-2o 8449  df-nadd 8648  df-no 27073  df-slt 27074  df-bday 27075  df-sle 27175  df-sslt 27209  df-scut 27211  df-0s 27251  df-made 27265  df-old 27266  df-left 27268  df-right 27269  df-norec 27338  df-norec2 27349  df-adds 27360  df-negs 27412  df-subs 27413  df-muls 27477

This theorem is referenced by:  mulscut  27501