Theorem addsproplem5 28007

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 28005	. . . 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 1144	. . 3 ⊢ (𝜑 → {(𝑋 +s 𝑌)} <<s ({𝑎 ∣ ∃𝑐 ∈ ( R ‘𝑋)𝑎 = (𝑐 +s 𝑌)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)}))
6		ovex 7465	. . . . 5 ⊢ (𝑋 +s 𝑌) ∈ V
7	6	snid 4661	. . . 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		bdayelon 27822	. . . . . . . . 9 ⊢ ( bday ‘𝑌) ∈ On
11		addspropord.4	. . . . . . . . 9 ⊢ (𝜑 → 𝑍 ∈ No )
12		oldbday 27940	. . . . . . . . 9 ⊢ ((( bday ‘𝑌) ∈ On ∧ 𝑍 ∈ No ) → (𝑍 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑍) ∈ ( bday ‘𝑌)))
13	10, 11, 12	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (𝑍 ∈ ( O ‘( bday ‘𝑌)) ↔ ( bday ‘𝑍) ∈ ( bday ‘𝑌)))
14	9, 13	mpbird 257	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ ( O ‘( bday ‘𝑌)))
15		addspropord.5	. . . . . . 7 ⊢ (𝜑 → 𝑌 <s 𝑍)
16		breq2 5146	. . . . . . . 8 ⊢ (𝑧 = 𝑍 → (𝑌 <s 𝑧 ↔ 𝑌 <s 𝑍))
17		rightval 27904	. . . . . . . 8 ⊢ ( R ‘𝑌) = {𝑧 ∈ ( O ‘( bday ‘𝑌)) ∣ 𝑌 <s 𝑧}
18	16, 17	elrab2 3694	. . . . . . 7 ⊢ (𝑍 ∈ ( R ‘𝑌) ↔ (𝑍 ∈ ( O ‘( bday ‘𝑌)) ∧ 𝑌 <s 𝑍))
19	14, 15, 18	sylanbrc 583	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ ( R ‘𝑌))
20		eqid 2736	. . . . . 6 ⊢ (𝑋 +s 𝑍) = (𝑋 +s 𝑍)
21		oveq2 7440	. . . . . . 7 ⊢ (𝑑 = 𝑍 → (𝑋 +s 𝑑) = (𝑋 +s 𝑍))
22	21	rspceeqv 3644	. . . . . 6 ⊢ ((𝑍 ∈ ( R ‘𝑌) ∧ (𝑋 +s 𝑍) = (𝑋 +s 𝑍)) → ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑))
23	19, 20, 22	sylancl 586	. . . . 5 ⊢ (𝜑 → ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑))
24		ovex 7465	. . . . . 6 ⊢ (𝑋 +s 𝑍) ∈ V
25		eqeq1 2740	. . . . . . 7 ⊢ (𝑏 = (𝑋 +s 𝑍) → (𝑏 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑍) = (𝑋 +s 𝑑)))
26	25	rexbidv 3178	. . . . . 6 ⊢ (𝑏 = (𝑋 +s 𝑍) → (∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑) ↔ ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑)))
27	24, 26	elab 3678	. . . . 5 ⊢ ((𝑋 +s 𝑍) ∈ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)} ↔ ∃𝑑 ∈ ( R ‘𝑌)(𝑋 +s 𝑍) = (𝑋 +s 𝑑))
28	23, 27	sylibr 234	. . . 4 ⊢ (𝜑 → (𝑋 +s 𝑍) ∈ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)})
29		elun2 4182	. . . 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)}))
30	28, 29	syl 17	. . 3 ⊢ (𝜑 → (𝑋 +s 𝑍) ∈ ({𝑎 ∣ ∃𝑐 ∈ ( R ‘𝑋)𝑎 = (𝑐 +s 𝑌)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( R ‘𝑌)𝑏 = (𝑋 +s 𝑑)}))
31	5, 8, 30	ssltsepcd 27840	. 2 ⊢ (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
32	3, 2	addscomd 28001	. 2 ⊢ (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
33	11, 2	addscomd 28001	. 2 ⊢ (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
34	31, 32, 33	3brtr4d 5174	1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  {cab 2713  ∀wral 3060  ∃wrex 3069   ∪ cun 3948  {csn 4625   class class class wbr 5142  Oncon0 6383  ‘cfv 6560  (class class class)co 7432   +no cnadd 8704   No csur 27685   <s cslt 27686   bday cbday 27687   <<s csslt 27826   O cold 27883   L cleft 27885   R cright 27886   +_s cadds 27993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-1o 8507  df-2o 8508  df-nadd 8705  df-no 27688  df-slt 27689  df-bday 27690  df-sslt 27827  df-scut 27829  df-0s 27870  df-made 27887  df-old 27888  df-left 27890  df-right 27891  df-norec2 27983  df-adds 27994

This theorem is referenced by:  addsproplem7  28009