Theorem mulcutlem 28523

Description: Lemma for mulcut 28524. 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 28522	. . . . . . . . 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 3205	. . . . . 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 3205	. . . . 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 3202	. . . 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 28517	. 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 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086   ∪ cun 4149  {csn 4584   class class class wbr 5102  ‘cfv 6539  (class class class)co 7421   +no cnadd 8678  Nocsur 28003   <s clts 28004  bdaycbday 28005   <<s cslts 28149   0_s c0s 28197  Lcleft 28217  Rcright 28218   +_s cadds 28351   −_s csubs 28412   ·_s cmuls 28498

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-1o 8480  df-2o 8481  df-nadd 8679  df-no 28006  df-lts 28007  df-bday 28008  df-les 28108  df-slts 28150  df-cuts 28152  df-0s 28199  df-made 28219  df-old 28220  df-left 28222  df-right 28223  df-norec 28330  df-norec2 28341  df-adds 28352  df-negs 28413  df-subs 28414  df-muls 28499

This theorem is used by:  mulcut  28524