Theorem mulcutlem 28361

Description: Lemma for mulcut 28362. State the theorem with extra DV conditions. (Contributed by Scott Fenton, 7-Mar-2025.)

Hypotheses

Ref	Expression
mulcutlem.1	⊢ (𝜑 → 𝐴 ∈ No )
mulcutlem.2	⊢ (𝜑 → 𝐵 ∈ No )

Assertion

Ref	Expression
mulcutlem	⊢ (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 mulcutlem

Dummy variables 𝑒 𝑓 𝑔 ℎ 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulcutlem.1	. 2 ⊢ (𝜑 → 𝐴 ∈ No )
2		mulcutlem.2	. 2 ⊢ (𝜑 → 𝐵 ∈ No )
3		mulsprop 28360	. . . . . . . . 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 26	. . . . . . . 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 1136	. . . . . . 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 3211	. . . . . 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 3211	. . . . 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 3208	. . . 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 488	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐴 ∈ No )
11		simpr 490	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ∈ No )
12	9, 10, 11	mulsproplem10 28355	. 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 596	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   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  {cab 2744  ∀wral 3082  ∃wrex 3092   ∪ cun 3906  {csn 4594   class class class wbr 5114  ‘cfv 6543  (class class class)co 7423   +no cnadd 8660   No csur 27841   <s clts 27842   bday cbday 27843   <<s cslts 27987   0_s c0s 28035   L cleft 28055   R cright 28056   +_s cadds 28189   -_s csubs 28250   ·_s cmuls 28336

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-1o 8462  df-2o 8463  df-nadd 8661  df-no 27844  df-lts 27845  df-bday 27846  df-les 27946  df-slts 27988  df-cuts 27990  df-0s 28037  df-made 28057  df-old 28058  df-left 28060  df-right 28061  df-norec 28168  df-norec2 28179  df-adds 28190  df-negs 28251  df-subs 28252  df-muls 28337

This theorem is used by:  mulcut  28362