Theorem mulcutlem 28550

Description: Lemma for mulcut 28551. 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 28549	. . . . . . . . 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 3209	. . . . . 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 3209	. . . . 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 3206	. . . 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 28544	. 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 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ∪ cun 4152  {csn 4588   class class class wbr 5106  ‘cfv 6561  (class class class)co 7446   +no cnadd 8705  Nocsur 28030   <s clts 28031  bdaycbday 28032   <<s cslts 28176   0_s c0s 28224  Lcleft 28244  Rcright 28245   +_s cadds 28378   −_s csubs 28439   ·_s cmuls 28525

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-1o 8507  df-2o 8508  df-nadd 8706  df-no 28033  df-lts 28034  df-bday 28035  df-les 28135  df-slts 28177  df-cuts 28179  df-0s 28226  df-made 28246  df-old 28247  df-left 28249  df-right 28250  df-norec 28357  df-norec2 28368  df-adds 28379  df-negs 28440  df-subs 28441  df-muls 28526

This theorem is used by:  mulcut  28551