Theorem addsproplem5 28341

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 28339	. . . 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 7445	. . . . 5 ⊢ (𝑋 +s 𝑌) ∈ V
7	6	snid 4623	. . . 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 28120	. . . . . . . . 9 ⊢ ( bday ‘𝑌) ∈ On
11		addspropord.4	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ No )
12		oldbday 28269	. . . . . . . . 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 28220	. . . . . . 7 ⊢ (𝑍 ∈ ( R ‘𝑌) ↔ (𝑍 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑌 <s 𝑍))
17	14, 15, 16	sylanbrc 595	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ ( R ‘𝑌))
18		eqid 2761	. . . . . 6 ⊢ (𝑋 +s 𝑍) = (𝑋 +s 𝑍)
19		oveq2 7420	. . . . . . 7 ⊢ (𝑑 = 𝑍 → (𝑋 +s 𝑑) = (𝑋 +s 𝑍))
20	19	rspceeqv 3599	. . . . . 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 7445	. . . . . 6 ⊢ (𝑋 +s 𝑍) ∈ V
23		eqeq1 2765	. . . . . . 7 ⊢ (𝑏 = (𝑋 +s 𝑍) → (𝑏 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑍) = (𝑋 +s 𝑑)))
24	23	rexbidv 3187	. . . . . 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 3633	. . . . 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 4129	. . . 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 28140	. 2 ⊢ (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
30	3, 2	addscomd 28335	. 2 ⊢ (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
31	11, 2	addscomd 28335	. 2 ⊢ (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
32	29, 30, 31	3brtr4d 5137	1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  ∃wrex 3087   ∪ cun 3897  {csn 4584   class class class wbr 5103  Oncon0 6355  ‘cfv 6531  (class class class)co 7412   +no cnadd 8658   No csur 27979   <s clts 27980   bday cbday 27981   <<s cslts 28125   O cold 28191   L cleft 28193   R cright 28194   +_s cadds 28327

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-1o 8460  df-2o 8461  df-nadd 8659  df-no 27982  df-lts 27983  df-bday 27984  df-slts 28126  df-cuts 28128  df-0s 28175  df-made 28195  df-old 28196  df-left 28198  df-right 28199  df-norec2 28317  df-adds 28328

This theorem is used by:  addsproplem7  28343