Theorem addsproplem4 28351

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 4105	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) = ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌)))
3	2	eleq2i 2853	. . . . . . . . 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 3109	. . . . . . 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 3138	. . . . . 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 28350	. . . 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 1161	. . 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 28131	. . . . . . . . 9 ⊢ ( bday ‘𝑍) ∈ On
14		addspropord.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ No )
15		oldbday 28280	. . . . . . . . 9 ⊢ ((( bday ‘𝑍) ∈ On ∧ 𝑌 ∈ No ) → (𝑌 ∈ ( O ‘( bday ‘𝑍)) ↔ ( bday ‘𝑌) ∈ ( bday ‘𝑍)))
16	13, 14, 15	sylancr 599	. . . . . . . 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 28230	. . . . . . 7 ⊢ (𝑌 ∈ ( L ‘𝑍) ↔ (𝑌 ∈ ( O ‘( bday ‘𝑍)) ∧ 𝑌 <s 𝑍))
20	17, 18, 19	sylanbrc 595	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ( L ‘𝑍))
21		eqid 2761	. . . . . 6 ⊢ (𝑋 +s 𝑌) = (𝑋 +s 𝑌)
22		oveq2 7426	. . . . . . 7 ⊢ (𝑑 = 𝑌 → (𝑋 +s 𝑑) = (𝑋 +s 𝑌))
23	22	rspceeqv 3599	. . . . . 6 ⊢ ((𝑌 ∈ ( L ‘𝑍) ∧ (𝑋 +s 𝑌) = (𝑋 +s 𝑌)) → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑌) = (𝑋 +s 𝑑))
24	20, 21, 23	sylancl 598	. . . . 5 ⊢ (𝜑 → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑌) = (𝑋 +s 𝑑))
25		ovex 7451	. . . . . 6 ⊢ (𝑋 +s 𝑌) ∈ V
26		eqeq1 2765	. . . . . . 7 ⊢ (𝑏 = (𝑋 +s 𝑌) → (𝑏 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑌) = (𝑋 +s 𝑑)))
27	26	rexbidv 3187	. . . . . 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 3633	. . . . 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 4129	. . . 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 7451	. . . . 5 ⊢ (𝑋 +s 𝑍) ∈ V
33	32	snid 4623	. . . 4 ⊢ (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)}
34	33	a1i 11	. . 3 ⊢ (𝜑 → (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)})
35	11, 31, 34	sltssepcd 28151	. 2 ⊢ (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
36	14, 8	addscomd 28346	. 2 ⊢ (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
37	9, 8	addscomd 28346	. 2 ⊢ (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
38	35, 36, 37	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 6361  ‘cfv 6537  (class class class)co 7418   +no cnadd 8667   No csur 27990   <s clts 27991   bday cbday 27992   <<s cslts 28136   O cold 28202   L cleft 28204   R cright 28205   +_s cadds 28338

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 7749

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 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-1o 8469  df-2o 8470  df-nadd 8668  df-no 27993  df-lts 27994  df-bday 27995  df-slts 28137  df-cuts 28139  df-0s 28186  df-made 28206  df-old 28207  df-left 28209  df-right 28210  df-norec2 28328  df-adds 28339

This theorem is used by:  addsproplem7  28354