Theorem addscutlem 27849

Description: Lemma for addscut 27850. Show the statement with some additional distinct variable conditions. (Contributed by Scott Fenton, 8-Mar-2025.)

Hypotheses

Ref	Expression
addscutlem.1	⊢ (𝜑 → 𝑋 ∈ No )
addscutlem.2	⊢ (𝜑 → 𝑌 ∈ No )

Assertion

Ref	Expression
addscutlem	⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})))

Distinct variable groups:   𝑋,𝑙,𝑚   𝑋,𝑝,𝑞,𝑟,𝑠   𝑡,𝑋,𝑠   𝑤,𝑋   𝑌,𝑙,𝑚   𝑌,𝑝,𝑞,𝑟,𝑠   𝑡,𝑌   𝑤,𝑌   𝑝,𝑙,𝜑,𝑞,𝑟,𝑠   𝜑,𝑚,𝑞,𝑟,𝑠   𝜑,𝑝,𝑞,𝑟,𝑠   𝑤,𝑝   𝜑,𝑡   𝜑,𝑤   𝑡,𝑞,𝑟   𝑤,𝑟

Proof of Theorem addscutlem

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addsprop 27848	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))
2	1	a1d 25	. . . 4 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ∧ 𝑧 ∈ No ) → (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0s ))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
3	2	rgen3 3196	. . 3 ⊢ ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0s ))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))
4	3	a1i 11	. 2 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘ 0s ))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
5		addscutlem.1	. 2 ⊢ (𝜑 → 𝑋 ∈ No )
6		addscutlem.2	. 2 ⊢ (𝜑 → 𝑌 ∈ No )
7	4, 5, 6	addsproplem3 27843	1 ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑝 ∣ ∃𝑙 ∈ ( L ‘𝑋)𝑝 = (𝑙 +s 𝑌)} ∪ {𝑞 ∣ ∃𝑚 ∈ ( L ‘𝑌)𝑞 = (𝑋 +s 𝑚)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑤 ∣ ∃𝑟 ∈ ( R ‘𝑋)𝑤 = (𝑟 +s 𝑌)} ∪ {𝑡 ∣ ∃𝑠 ∈ ( R ‘𝑌)𝑡 = (𝑋 +s 𝑠)})))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2703  ∀wral 3055  ∃wrex 3064   ∪ cun 3941  {csn 4623   class class class wbr 5141  ‘cfv 6537  (class class class)co 7405   +no cnadd 8666   No csur 27528   <s cslt 27529   bday cbday 27530   <<s csslt 27668   0_s c0s 27710   L cleft 27727   R cright 27728   +_s cadds 27831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  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 6294  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-1o 8467  df-2o 8468  df-nadd 8667  df-no 27531  df-slt 27532  df-bday 27533  df-sslt 27669  df-scut 27671  df-0s 27712  df-made 27729  df-old 27730  df-left 27732  df-right 27733  df-norec2 27821  df-adds 27832

This theorem is referenced by:  addscut  27850