Theorem addsproplem4 27910

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 4103	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) = ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌)))
3	2	eleq2i 2823	. . . . . . . . 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 349	. . . . . . . 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 3078	. . . . . . 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 3107	. . . . . 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 218	. . . . 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 27909	. . . 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 1143	. . 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		bdayelon 27710	. . . . . . . . 9 ⊢ ( bday ‘𝑍) ∈ On
14		addspropord.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ No )
15		oldbday 27841	. . . . . . . . 9 ⊢ ((( bday ‘𝑍) ∈ On ∧ 𝑌 ∈ No ) → (𝑌 ∈ ( O ‘( bday ‘𝑍)) ↔ ( bday ‘𝑌) ∈ ( bday ‘𝑍)))
16	13, 14, 15	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (𝑌 ∈ ( O ‘( bday ‘𝑍)) ↔ ( bday ‘𝑌) ∈ ( bday ‘𝑍)))
17	12, 16	mpbird 257	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ ( O ‘( bday ‘𝑍)))
18		addspropord.5	. . . . . . 7 ⊢ (𝜑 → 𝑌 <s 𝑍)
19		elleft 27801	. . . . . . 7 ⊢ (𝑌 ∈ ( L ‘𝑍) ↔ (𝑌 ∈ ( O ‘( bday ‘𝑍)) ∧ 𝑌 <s 𝑍))
20	17, 18, 19	sylanbrc 583	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ( L ‘𝑍))
21		eqid 2731	. . . . . 6 ⊢ (𝑋 +s 𝑌) = (𝑋 +s 𝑌)
22		oveq2 7349	. . . . . . 7 ⊢ (𝑑 = 𝑌 → (𝑋 +s 𝑑) = (𝑋 +s 𝑌))
23	22	rspceeqv 3595	. . . . . 6 ⊢ ((𝑌 ∈ ( L ‘𝑍) ∧ (𝑋 +s 𝑌) = (𝑋 +s 𝑌)) → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑌) = (𝑋 +s 𝑑))
24	20, 21, 23	sylancl 586	. . . . 5 ⊢ (𝜑 → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑌) = (𝑋 +s 𝑑))
25		ovex 7374	. . . . . 6 ⊢ (𝑋 +s 𝑌) ∈ V
26		eqeq1 2735	. . . . . . 7 ⊢ (𝑏 = (𝑋 +s 𝑌) → (𝑏 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑌) = (𝑋 +s 𝑑)))
27	26	rexbidv 3156	. . . . . 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 3630	. . . . 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 234	. . . 4 ⊢ (𝜑 → (𝑋 +s 𝑌) ∈ {𝑏 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑏 = (𝑋 +s 𝑑)})
30		elun2 4128	. . . 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 17	. . 3 ⊢ (𝜑 → (𝑋 +s 𝑌) ∈ ({𝑎 ∣ ∃𝑐 ∈ ( L ‘𝑋)𝑎 = (𝑐 +s 𝑍)} ∪ {𝑏 ∣ ∃𝑑 ∈ ( L ‘𝑍)𝑏 = (𝑋 +s 𝑑)}))
32		ovex 7374	. . . . 5 ⊢ (𝑋 +s 𝑍) ∈ V
33	32	snid 4610	. . . 4 ⊢ (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)}
34	33	a1i 11	. . 3 ⊢ (𝜑 → (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)})
35	11, 31, 34	ssltsepcd 27730	. 2 ⊢ (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
36	14, 8	addscomd 27905	. 2 ⊢ (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
37	9, 8	addscomd 27905	. 2 ⊢ (𝜑 → (𝑍 +s 𝑋) = (𝑋 +s 𝑍))
38	35, 36, 37	3brtr4d 5118	1 ⊢ (𝜑 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056   ∪ cun 3895  {csn 4571   class class class wbr 5086  Oncon0 6301  ‘cfv 6476  (class class class)co 7341   +no cnadd 8575   No csur 27573   <s cslt 27574   bday cbday 27575   <<s csslt 27715   O cold 27779   L cleft 27781   R cright 27782   +_s cadds 27897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-se 5565  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-1o 8380  df-2o 8381  df-nadd 8576  df-no 27576  df-slt 27577  df-bday 27578  df-sslt 27716  df-scut 27718  df-0s 27763  df-made 27783  df-old 27784  df-left 27786  df-right 27787  df-norec2 27887  df-adds 27898

This theorem is referenced by:  addsproplem7  27913