Theorem addscutlem 27461

Description: Lemma for addscut 27462. 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 27460	. . . . 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 3203	. . 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 27455	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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  ∀wral 3062  ∃wrex 3071   ∪ cun 3947  {csn 4629   class class class wbr 5149  ‘cfv 6544  (class class class)co 7409   +no cnadd 8664   No csur 27143   <s cslt 27144   bday cbday 27145   <<s csslt 27282   0_s c0s 27323   L cleft 27340   R cright 27341   +_s cadds 27443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-1o 8466  df-2o 8467  df-nadd 8665  df-no 27146  df-slt 27147  df-bday 27148  df-sslt 27283  df-scut 27285  df-0s 27325  df-made 27342  df-old 27343  df-left 27345  df-right 27346  df-norec2 27433  df-adds 27444

This theorem is referenced by:  addscut  27462