Theorem addsproplem1 28412

Description: Lemma for surreal addition properties. To prove closure on surreal addition we need to prove that addition is compatible with order at the same time. We do this by inducting over the maximum of two natural sums of the birthdays of surreals numbers. In the final step we will loop around and use tfr3 8445 to prove this of all surreals. This first lemma just instantiates the inductive hypothesis so we do not need to do it continuously throughout the proof. (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)))))
addsproplem1.2	⊢ (𝜑 → 𝐴 ∈ No)
addsproplem1.3	⊢ (𝜑 → 𝐵 ∈ No)
addsproplem1.4	⊢ (𝜑 → 𝐶 ∈ No)
addsproplem1.5	⊢ (𝜑 → (((bday‘𝐴) +no (bday‘𝐵)) ∪ ((bday‘𝐴) +no (bday‘𝐶))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))))

Assertion

Ref	Expression
addsproplem1	⊢ (𝜑 → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧   𝑧,𝐶   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)   𝐵(𝑥)   𝐶(𝑥, 𝑦)

Proof of Theorem addsproplem1

Step	Hyp	Ref	Expression
1		addsproplem1.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ No)
2		addsproplem1.3	. . 3 ⊢ (𝜑 → 𝐵 ∈ No)
3		addsproplem1.4	. . 3 ⊢ (𝜑 → 𝐶 ∈ No)
4	1, 2, 3	3jca 1146	. 2 ⊢ (𝜑 → (𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No))
5		addsproplem.1	. 2 ⊢ (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)))))
6		addsproplem1.5	. 2 ⊢ (𝜑 → (((bday‘𝐴) +no (bday‘𝐵)) ∪ ((bday‘𝐴) +no (bday‘𝐶))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))))
7		fveq2 6916	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (bday‘𝑥) = (bday‘𝐴))
8	7	oveq1d 7466	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((bday‘𝑥) +no (bday‘𝑦)) = ((bday‘𝐴) +no (bday‘𝑦)))
9	7	oveq1d 7466	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((bday‘𝑥) +no (bday‘𝑧)) = ((bday‘𝐴) +no (bday‘𝑧)))
10	8, 9	uneq12d 4171	. . . . 5 ⊢ (𝑥 = 𝐴 → (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) = (((bday‘𝐴) +no (bday‘𝑦)) ∪ ((bday‘𝐴) +no (bday‘𝑧))))
11	10	eleq1d 2849	. . . 4 ⊢ (𝑥 = 𝐴 → ((((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) ↔ (((bday‘𝐴) +no (bday‘𝑦)) ∪ ((bday‘𝐴) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍)))))
12		oveq1 7458	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
13	12	eleq1d 2849	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 +s 𝑦) ∈ No ↔ (𝐴 +s 𝑦) ∈ No))
14		oveq2 7459	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 +s 𝑥) = (𝑦 +s 𝐴))
15		oveq2 7459	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑧 +s 𝑥) = (𝑧 +s 𝐴))
16	14, 15	breq12d 5121	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑦 +s 𝑥) <s (𝑧 +s 𝑥) ↔ (𝑦 +s 𝐴) <s (𝑧 +s 𝐴)))
17	16	imbi2d 343	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)) ↔ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴))))
18	13, 17	anbi12d 644	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))) ↔ ((𝐴 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴)))))
19	11, 18	imbi12d 347	. . 3 ⊢ (𝑥 = 𝐴 → (((((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‘𝐴) +no (bday‘𝑦)) ∪ ((bday‘𝐴) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) → ((𝐴 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴))))))
20		fveq2 6916	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (bday‘𝑦) = (bday‘𝐵))
21	20	oveq2d 7467	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((bday‘𝐴) +no (bday‘𝑦)) = ((bday‘𝐴) +no (bday‘𝐵)))
22	21	uneq1d 4169	. . . . 5 ⊢ (𝑦 = 𝐵 → (((bday‘𝐴) +no (bday‘𝑦)) ∪ ((bday‘𝐴) +no (bday‘𝑧))) = (((bday‘𝐴) +no (bday‘𝐵)) ∪ ((bday‘𝐴) +no (bday‘𝑧))))
23	22	eleq1d 2849	. . . 4 ⊢ (𝑦 = 𝐵 → ((((bday‘𝐴) +no (bday‘𝑦)) ∪ ((bday‘𝐴) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) ↔ (((bday‘𝐴) +no (bday‘𝐵)) ∪ ((bday‘𝐴) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍)))))
24		oveq2 7459	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
25	24	eleq1d 2849	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴 +s 𝑦) ∈ No ↔ (𝐴 +s 𝐵) ∈ No))
26		breq1 5111	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 <s 𝑧 ↔ 𝐵 <s 𝑧))
27		oveq1 7458	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝐴) = (𝐵 +s 𝐴))
28	27	breq1d 5118	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑦 +s 𝐴) <s (𝑧 +s 𝐴) ↔ (𝐵 +s 𝐴) <s (𝑧 +s 𝐴)))
29	26, 28	imbi12d 347	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴)) ↔ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴))))
30	25, 29	anbi12d 644	. . . 4 ⊢ (𝑦 = 𝐵 → (((𝐴 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴))) ↔ ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴)))))
31	23, 30	imbi12d 347	. . 3 ⊢ (𝑦 = 𝐵 → (((((bday‘𝐴) +no (bday‘𝑦)) ∪ ((bday‘𝐴) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) → ((𝐴 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝐴) <s (𝑧 +s 𝐴)))) ↔ ((((bday‘𝐴) +no (bday‘𝐵)) ∪ ((bday‘𝐴) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴))))))
32		fveq2 6916	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (bday‘𝑧) = (bday‘𝐶))
33	32	oveq2d 7467	. . . . . 6 ⊢ (𝑧 = 𝐶 → ((bday‘𝐴) +no (bday‘𝑧)) = ((bday‘𝐴) +no (bday‘𝐶)))
34	33	uneq2d 4170	. . . . 5 ⊢ (𝑧 = 𝐶 → (((bday‘𝐴) +no (bday‘𝐵)) ∪ ((bday‘𝐴) +no (bday‘𝑧))) = (((bday‘𝐴) +no (bday‘𝐵)) ∪ ((bday‘𝐴) +no (bday‘𝐶))))
35	34	eleq1d 2849	. . . 4 ⊢ (𝑧 = 𝐶 → ((((bday‘𝐴) +no (bday‘𝐵)) ∪ ((bday‘𝐴) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) ↔ (((bday‘𝐴) +no (bday‘𝐵)) ∪ ((bday‘𝐴) +no (bday‘𝐶))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍)))))
36		breq2 5112	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝐵 <s 𝑧 ↔ 𝐵 <s 𝐶))
37		oveq1 7458	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑧 +s 𝐴) = (𝐶 +s 𝐴))
38	37	breq2d 5120	. . . . . 6 ⊢ (𝑧 = 𝐶 → ((𝐵 +s 𝐴) <s (𝑧 +s 𝐴) ↔ (𝐵 +s 𝐴) <s (𝐶 +s 𝐴)))
39	36, 38	imbi12d 347	. . . . 5 ⊢ (𝑧 = 𝐶 → ((𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴)) ↔ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))
40	39	anbi2d 642	. . . 4 ⊢ (𝑧 = 𝐶 → (((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴))) ↔ ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴)))))
41	35, 40	imbi12d 347	. . 3 ⊢ (𝑧 = 𝐶 → (((((bday‘𝐴) +no (bday‘𝐵)) ∪ ((bday‘𝐴) +no (bday‘𝑧))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝑧 → (𝐵 +s 𝐴) <s (𝑧 +s 𝐴)))) ↔ ((((bday‘𝐴) +no (bday‘𝐵)) ∪ ((bday‘𝐴) +no (bday‘𝐶))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))))
42	19, 31, 41	rspc3v 3595	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐶 ∈ No) → (∀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)))) → ((((bday‘𝐴) +no (bday‘𝐵)) ∪ ((bday‘𝐴) +no (bday‘𝐶))) ∈ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))))
43	4, 5, 6, 42	syl3c 67	1 ⊢ (𝜑 → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ∪ cun 4152   class class class wbr 5108  ‘cfv 6566  (class class class)co 7451   +no cnadd 8712  Nocsur 28054   <s clts 28055  bdaycbday 28056   +_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-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574  df-ov 7454

This theorem is used by:  addsproplem2  28413  addsproplem6  28417