Theorem addsproplem4 27978

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 4099	. . . . . . . . . 10 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) = ((( bday ‘𝑋) +no ( bday ‘𝑍)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑌)))
3	2	eleq2i 2829	. . . . . . . . 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 3084	. . . . . . 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 3113	. . . . . 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 27977	. . . 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 1144	. . 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 27758	. . . . . . . . 9 ⊢ ( bday ‘𝑍) ∈ On
14		addspropord.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ No )
15		oldbday 27907	. . . . . . . . 9 ⊢ ((( bday ‘𝑍) ∈ On ∧ 𝑌 ∈ No ) → (𝑌 ∈ ( O ‘( bday ‘𝑍)) ↔ ( bday ‘𝑌) ∈ ( bday ‘𝑍)))
16	13, 14, 15	sylancr 588	. . . . . . . 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 27857	. . . . . . 7 ⊢ (𝑌 ∈ ( L ‘𝑍) ↔ (𝑌 ∈ ( O ‘( bday ‘𝑍)) ∧ 𝑌 <s 𝑍))
20	17, 18, 19	sylanbrc 584	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ( L ‘𝑍))
21		eqid 2737	. . . . . 6 ⊢ (𝑋 +s 𝑌) = (𝑋 +s 𝑌)
22		oveq2 7368	. . . . . . 7 ⊢ (𝑑 = 𝑌 → (𝑋 +s 𝑑) = (𝑋 +s 𝑌))
23	22	rspceeqv 3588	. . . . . 6 ⊢ ((𝑌 ∈ ( L ‘𝑍) ∧ (𝑋 +s 𝑌) = (𝑋 +s 𝑌)) → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑌) = (𝑋 +s 𝑑))
24	20, 21, 23	sylancl 587	. . . . 5 ⊢ (𝜑 → ∃𝑑 ∈ ( L ‘𝑍)(𝑋 +s 𝑌) = (𝑋 +s 𝑑))
25		ovex 7393	. . . . . 6 ⊢ (𝑋 +s 𝑌) ∈ V
26		eqeq1 2741	. . . . . . 7 ⊢ (𝑏 = (𝑋 +s 𝑌) → (𝑏 = (𝑋 +s 𝑑) ↔ (𝑋 +s 𝑌) = (𝑋 +s 𝑑)))
27	26	rexbidv 3162	. . . . . 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 3623	. . . . 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 4124	. . . 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 7393	. . . . 5 ⊢ (𝑋 +s 𝑍) ∈ V
33	32	snid 4607	. . . 4 ⊢ (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)}
34	33	a1i 11	. . 3 ⊢ (𝜑 → (𝑋 +s 𝑍) ∈ {(𝑋 +s 𝑍)})
35	11, 31, 34	sltssepcd 27778	. 2 ⊢ (𝜑 → (𝑋 +s 𝑌) <s (𝑋 +s 𝑍))
36	14, 8	addscomd 27973	. 2 ⊢ (𝜑 → (𝑌 +s 𝑋) = (𝑋 +s 𝑌))
37	9, 8	addscomd 27973	. 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 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062   ∪ cun 3888  {csn 4568   class class class wbr 5086  Oncon0 6317  ‘cfv 6492  (class class class)co 7360   +no cnadd 8594   No csur 27617   <s clts 27618   bday cbday 27619   <<s cslts 27763   O cold 27829   L cleft 27831   R cright 27832   +_s cadds 27965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-1o 8398  df-2o 8399  df-nadd 8595  df-no 27620  df-lts 27621  df-bday 27622  df-slts 27764  df-cuts 27766  df-0s 27813  df-made 27833  df-old 27834  df-left 27836  df-right 27837  df-norec2 27955  df-adds 27966

This theorem is referenced by:  addsproplem7  27981