Theorem addsproplem4 28127

Description: Lemma for surreal addition properties. Show the second half of the inductive hypothesis when 𝑌 is older than 𝑍. (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)
addsproplem4.6	⊢ (𝜑 → ( bday ‘𝑌) ∈ ( bday ‘𝑍))

Assertion

Ref	Expression
addsproplem4	⊢ (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 addsproplem4

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	. . . . . 6 ⊢ (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		uncom 4120	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) = ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌)))
3	2	eleq2i 2861	. . . . . . . . 9 ⊢ (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) ↔ ((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))))
4	3	imbi1i 352	. . . . . . . 8 ⊢ ((((( 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)))) ↔ (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
5	4	ralbii 3117	. . . . . . 7 ⊢ (∀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)))) ↔ ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
6	5	2ralbii 3146	. . . . . 6 ⊢ (∀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)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
7	1, 6	sylib 221	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (((( bday ‘𝑥) +no ( bday ‘𝑦)) ∪ (( bday ‘𝑥) +no ( bday ‘𝑧))) ∈ ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
8		addspropord.2	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ No )
9		addspropord.4	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ No )
10	7, 8, 9	addsproplem3 28126	. . . 4 ⊢ (𝜑 → ((𝑋 +s 𝑍) ∈ No ∧ ({𝑎 ∣ ∃𝑐 ∈ ( L ‘𝑋)𝑎 = (𝑐 +s 𝑍)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑏 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)} ∧ {(𝑋 +s 𝑍)} <<s ({𝑒 ∣ ∃𝑔 ∈ ( R ‘𝑋)𝑒 = (𝑔 +s 𝑍)} ∪ {𝑓 ∣ ∃ℎ ∈ ( R ‘𝑍)𝑓 = (𝑋 +s ℎ)})))
11	10	simp2d 1159	. . 3 ⊢ (𝜑 → ({𝑎 ∣ ∃𝑐 ∈ ( L ‘𝑋)𝑎 = (𝑐 +s 𝑍)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑏 = (𝑋 +s 𝑑)}) <<s {(𝑋 +s 𝑍)})
12		addsproplem4.6	. . . . . . . 8 ⊢ (𝜑 → ( bday ‘𝑌) ∈ ( bday ‘𝑍))
13		bdayon 27907	. . . . . . . . 9 ⊢ ( bday ‘𝑍) ∈ On
14		addspropord.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ No )
15		oldbday 28056	. . . . . . . . 9 ⊢ ((( bday ‘𝑍) ∈ On ∧ 𝑌 ∈ No ) → (𝑌 ∈ ( O ‘( bday ‘𝑍)) ↔ ( bday ‘𝑌) ∈ ( bday ‘𝑍)))
16	13, 14, 15	sylancr 598	. . . . . . . 8 ⊢ (𝜑 → (𝑌 ∈ ( O ‘( bday ‘𝑍)) ↔ ( bday ‘𝑌) ∈ ( bday ‘𝑍)))
17	12, 16	mpbird 260	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ ( O ‘( bday ‘𝑍)))
18		addspropord.5	. . . . . . 7 ⊢ (𝜑 → 𝑌 <s 𝑍)
19		elleft 28006	. . . . . . 7 ⊢ (𝑌 ∈ ( L ‘𝑍) ↔ (𝑌 ∈ ( O ‘( bday ‘𝑍)) ∧ 𝑌 <s 𝑍))
20	17, 18, 19	sylanbrc 594	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ( L ‘𝑍))
21		eqid 2769	. . . . . 6 ⊢ (𝑋 +s 𝑌) = (𝑋 +s 𝑌)
22		oveq2 7416	. . . . . . 7 ⊢ (𝑑 = 𝑌 → (𝑋 +s 𝑑) = (𝑋 +s 𝑌))
23	22	rspceeqv 3613	. . . . . 6 ⊢ ((𝑌 ∈ ( L ‘𝑍) ∧ (𝑋 +s 𝑌) = (𝑋 +s 𝑌)) → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑌) = (𝑋 +s 𝑑))
24	20, 21, 23	sylancl 597	. . . . 5 ⊢ (𝜑 → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑌) = (𝑋 +s 𝑑))
25		ovex 7441	. . . . . 6 ⊢ (𝑋 +s 𝑌) ∈ V
26		eqeq1 2773	. . . . . . 7 ⊢ (𝑏 = (𝑋 +s 𝑌) → (𝑏 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑌) = (𝑋 +s 𝑑)))
27	26	rexbidv 3195	. . . . . 6 ⊢ (𝑏 = (𝑋 +s 𝑌) → (∃𝑑 ∈ ( L ‘𝑍)𝑏 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑌) = (𝑋 +s 𝑑)))
28	25, 27	elab 3647	. . . . 5 ⊢ ((𝑋 +s 𝑌) ∈ {𝑏 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑏 = (𝑋 +s 𝑑)} ↔ ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑌) = (𝑋 +s 𝑑))
29	24, 28	sylibr 237	. . . 4 ⊢ (𝜑 → (𝑋 +s 𝑌) ∈ {𝑏 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑏 = (𝑋 +s 𝑑)})
30		elun2 4144	. . . 4 ⊢ ((𝑋 +s 𝑌) ∈ {𝑏 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑏 = (𝑋 +s 𝑑)} → (𝑋 +s 𝑌) ∈ ({𝑎 ∣ ∃𝑐 ∈ ( L ‘𝑋)𝑎 = (𝑐 +s 𝑍)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑏 = (𝑋 +s 𝑑)}))
31	29, 30	syl 18	. . 3 ⊢ (𝜑 → (𝑋 +s 𝑌) ∈ ({𝑎 ∣ ∃𝑐 ∈ ( L ‘𝑋)𝑎 = (𝑐 +s 𝑍)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑏 = (𝑋 +s 𝑑)}))
32		ovex 7441	. . . . 5 ⊢ (𝑋 +s 𝑍) ∈ V
33	32	snid 4630	. . . 4 ⊢ (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)}
34	33	a1i 11	. . 3 ⊢ (𝜑 → (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)})
35	11, 31, 34	sltssepcd 27927	. 2 ⊢ (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
36	14, 8	addscomd 28122	. 2 ⊢ (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
37	9, 8	addscomd 28122	. 2 ⊢ (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
38	35, 36, 37	3brtr4d 5144	1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085  ∃wrex 3095   ∪ cun 3911  {csn 4591   class class class wbr 5110  Oncon0 6358  ‘cfv 6534  (class class class)co 7408   +no cnadd 8647   No csur 27766   <s clts 27767   bday cbday 27768   <<s cslts 27912   O cold 27978   L cleft 27980   R cright 27981   +_s cadds 28114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-1o 8449  df-2o 8450  df-nadd 8648  df-no 27769  df-lts 27770  df-bday 27771  df-slts 27913  df-cuts 27915  df-0s 27962  df-made 27982  df-old 27983  df-left 27985  df-right 27986  df-norec2 28104  df-adds 28115

This theorem is referenced by:  addsproplem7  28130