Theorem mulcutlem 28201

Description: Lemma for mulcut 28202. 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 28200	. . . . . . . . 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 1130	. . . . . . 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 3204	. . . . . 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 3204	. . . . 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 3201	. . . 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 486	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐴 ∈ No )
11		simpr 488	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → 𝐵 ∈ No )
12	9, 10, 11	mulsproplem10 28195	. 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 593	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 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  {cab 2739  ∀wral 3075  ∃wrex 3085   ∪ cun 3902  {csn 4581   class class class wbr 5099  ‘cfv 6517  (class class class)co 7392   +no cnadd 8630   No csur 27681   <s clts 27682   bday cbday 27683   <<s cslts 27827   0_s c0s 27875   L cleft 27895   R cright 27896   +_s cadds 28029   -_s csubs 28090   ·_s cmuls 28176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-ot 4590  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-1o 8432  df-2o 8433  df-nadd 8631  df-no 27684  df-lts 27685  df-bday 27686  df-les 27786  df-slts 27828  df-cuts 27830  df-0s 27877  df-made 27897  df-old 27898  df-left 27900  df-right 27901  df-norec 28008  df-norec2 28019  df-adds 28030  df-negs 28091  df-subs 28092  df-muls 28177

This theorem is referenced by:  mulcut  28202