Theorem addsproplem4 28415

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 4160	. . . . . . . . . 10 ⊢ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) = (((bday‘𝑋) +no (bday‘𝑍)) ∪ ((bday‘𝑋) +no (bday‘𝑌)))
3	2	eleq2i 2856	. . . . . . . . 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 3112	. . . . . . 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 3141	. . . . . 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 28414	. . . 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 28195	. . . . . . . . 9 ⊢ (bday‘𝑍) ∈ On
14		addspropord.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ No)
15		oldbday 28344	. . . . . . . . 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 28294	. . . . . . 7 ⊢ (𝑌 ∈ (L‘𝑍) ↔ (𝑌 ∈ (O‘(bday‘𝑍)) ∧ 𝑌 <s 𝑍))
20	17, 18, 19	sylanbrc 595	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (L‘𝑍))
21		eqid 2763	. . . . . 6 ⊢ (𝑋 +s 𝑌) = (𝑋 +s 𝑌)
22		oveq2 7459	. . . . . . 7 ⊢ (𝑑 = 𝑌 → (𝑋 +s 𝑑) = (𝑋 +s 𝑌))
23	22	rspceeqv 3602	. . . . . 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 7484	. . . . . 6 ⊢ (𝑋 +s 𝑌) ∈ V
26		eqeq1 2767	. . . . . . 7 ⊢ (𝑏 = (𝑋 +s 𝑌) → (𝑏 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑌) = (𝑋 +s 𝑑)))
27	26	rexbidv 3190	. . . . . 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 3636	. . . . 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 4184	. . . 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 7484	. . . . 5 ⊢ (𝑋 +s 𝑍) ∈ V
33	32	snid 4624	. . . 4 ⊢ (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)}
34	33	a1i 11	. . 3 ⊢ (𝜑 → (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)})
35	11, 31, 34	sltssepcd 28215	. 2 ⊢ (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
36	14, 8	addscomd 28410	. 2 ⊢ (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
37	9, 8	addscomd 28410	. 2 ⊢ (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
38	35, 36, 37	3brtr4d 5143	1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ∪ cun 4152  {csn 4588   class class class wbr 5108  Oncon0 6388  ‘cfv 6566  (class class class)co 7451   +no cnadd 8712  Nocsur 28054   <s clts 28055  bdaycbday 28056   <<s cslts 28200  Ocold 28266  Lcleft 28268  Rcright 28269   +_s cadds 28402

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-1o 8514  df-2o 8515  df-nadd 8713  df-no 28057  df-lts 28058  df-bday 28059  df-slts 28201  df-cuts 28203  df-0s 28250  df-made 28270  df-old 28271  df-left 28273  df-right 28274  df-norec2 28392  df-adds 28403

This theorem is used by:  addsproplem7  28418