Theorem addsproplem5 28203

Description: Lemma for surreal addition properties. Show the second half of the inductive hypothesis when 𝑍 is older than 𝑌. (Contributed by Scott Fenton, 21-Jan-2025.)

Hypotheses

Ref	Expression
addsproplem.1	⊢ (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 ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
addspropord.2	⊢ (𝜑 → 𝑋 ∈ No )
addspropord.3	⊢ (𝜑 → 𝑌 ∈ No )
addspropord.4	⊢ (𝜑 → 𝑍 ∈ No )
addspropord.5	⊢ (𝜑 → 𝑌 <s 𝑍)
addsproplem5.6	⊢ (𝜑 → ( bday ‘𝑍) ∈ ( bday ‘𝑌))

Assertion

Ref	Expression
addsproplem5	⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

Distinct variable groups:   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)

Proof of Theorem addsproplem5

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addsproplem.1	. . . . 5 ⊢ (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 ‘𝑍))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
2		addspropord.2	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ No )
3		addspropord.3	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ No )
4	1, 2, 3	addsproplem3 28201	. . . 4 ⊢ (𝜑 → ((𝑋 +s 𝑌) ∈ No ∧ ({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝑋)𝑒 = (𝑓 +s 𝑌)} ∪ {𝑔 ∣ ∃ℎ ∈ ( L ‘𝑌)𝑔 = (𝑋 +s ℎ)}) <<s {(𝑋 +s 𝑌)} ∧ {(𝑋 +s 𝑌)} <<s ({𝑎 ∣ ∃𝑐 ∈ ( R ‘𝑋)𝑎 = (𝑐 +s 𝑌)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)})))
5	4	simp3d 1162	. . 3 ⊢ (𝜑 → {(𝑋 +s 𝑌)} <<s ({𝑎 ∣ ∃𝑐 ∈ ( R ‘𝑋)𝑎 = (𝑐 +s 𝑌)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)}))
6		ovex 7456	. . . . 5 ⊢ (𝑋 +s 𝑌) ∈ V
7	6	snid 4633	. . . 4 ⊢ (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)}
8	7	a1i 11	. . 3 ⊢ (𝜑 → (𝑋 +s 𝑌) ∈ {(𝑋 +s 𝑌)})
9		addsproplem5.6	. . . . . . . 8 ⊢ (𝜑 → ( bday ‘𝑍) ∈ ( bday ‘𝑌))
10		bdayon 27982	. . . . . . . . 9 ⊢ ( bday ‘𝑌) ∈ On
11		addspropord.4	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ No )
12		oldbday 28131	. . . . . . . . 9 ⊢ ((( bday ‘𝑌) ∈ On ∧ 𝑍 ∈ No ) → (𝑍 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑍) ∈ ( bday ‘𝑌)))
13	10, 11, 12	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → (𝑍 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑍) ∈ ( bday ‘𝑌)))
14	9, 13	mpbird 260	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ ( O ‘( bday ‘𝑌)))
15		addspropord.5	. . . . . . 7 ⊢ (𝜑 → 𝑌 <s 𝑍)
16		elright 28082	. . . . . . 7 ⊢ (𝑍 ∈ ( R ‘𝑌) ↔ (𝑍 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑌 <s 𝑍))
17	14, 15, 16	sylanbrc 595	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ ( R ‘𝑌))
18		eqid 2766	. . . . . 6 ⊢ (𝑋 +s 𝑍) = (𝑋 +s 𝑍)
19		oveq2 7431	. . . . . . 7 ⊢ (𝑑 = 𝑍 → (𝑋 +s 𝑑) = (𝑋 +s 𝑍))
20	19	rspceeqv 3607	. . . . . 6 ⊢ ((𝑍 ∈ ( R ‘𝑌) ∧ (𝑋 +s 𝑍) = (𝑋 +s 𝑍)) → ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑))
21	17, 18, 20	sylancl 598	. . . . 5 ⊢ (𝜑 → ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑))
22		ovex 7456	. . . . . 6 ⊢ (𝑋 +s 𝑍) ∈ V
23		eqeq1 2770	. . . . . . 7 ⊢ (𝑏 = (𝑋 +s 𝑍) → (𝑏 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑍) = (𝑋 +s 𝑑)))
24	23	rexbidv 3192	. . . . . 6 ⊢ (𝑏 = (𝑋 +s 𝑍) → (∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑)))
25	22, 24	elab 3641	. . . . 5 ⊢ ((𝑋 +s 𝑍) ∈ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)} ↔ ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑))
26	21, 25	sylibr 237	. . . 4 ⊢ (𝜑 → (𝑋 +s 𝑍) ∈ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)})
27		elun2 4139	. . . 4 ⊢ ((𝑋 +s 𝑍) ∈ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)} → (𝑋 +s 𝑍) ∈ ({𝑎 ∣ ∃𝑐 ∈ ( R ‘𝑋)𝑎 = (𝑐 +s 𝑌)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)}))
28	26, 27	syl 18	. . 3 ⊢ (𝜑 → (𝑋 +s 𝑍) ∈ ({𝑎 ∣ ∃𝑐 ∈ ( R ‘𝑋)𝑎 = (𝑐 +s 𝑌)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)}))
29	5, 8, 28	sltssepcd 28002	. 2 ⊢ (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
30	3, 2	addscomd 28197	. 2 ⊢ (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
31	11, 2	addscomd 28197	. 2 ⊢ (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
32	29, 30, 31	3brtr4d 5148	1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {cab 2744  ∀wral 3082  ∃wrex 3092   ∪ cun 3906  {csn 4594   class class class wbr 5114  Oncon0 6367  ‘cfv 6543  (class class class)co 7423   +no cnadd 8660   No csur 27841   <s clts 27842   bday cbday 27843   <<s cslts 27987   O cold 28053   L cleft 28055   R cright 28056   +_s cadds 28189

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-1o 8462  df-2o 8463  df-nadd 8661  df-no 27844  df-lts 27845  df-bday 27846  df-slts 27988  df-cuts 27990  df-0s 28037  df-made 28057  df-old 28058  df-left 28060  df-right 28061  df-norec2 28179  df-adds 28190

This theorem is used by:  addsproplem7  28205