Theorem addsproplem1 28348

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 8400 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 6883	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ( bday ‘𝑥) = ( bday ‘𝐴))
8	7	oveq1d 7433	. . . . . 6 ⊢ (𝑥 = 𝐴 → (( bday ‘𝑥) +no ( bday ‘𝑦)) = (( bday ‘𝐴) +no ( bday ‘𝑦)))
9	7	oveq1d 7433	. . . . . 6 ⊢ (𝑥 = 𝐴 → (( bday ‘𝑥) +no ( bday ‘𝑧)) = (( bday ‘𝐴) +no ( bday ‘𝑧)))
10	8, 9	uneq12d 4116	. . . . 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 2846	. . . 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 7425	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
13	12	eleq1d 2846	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 +s 𝑦) ∈ No ↔ (𝐴 +s 𝑦) ∈ No ))
14		oveq2 7426	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 +s 𝑥) = (𝑦 +s 𝐴))
15		oveq2 7426	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑧 +s 𝑥) = (𝑧 +s 𝐴))
16	14, 15	breq12d 5116	. . . . . 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 6883	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ( bday ‘𝑦) = ( bday ‘𝐵))
21	20	oveq2d 7434	. . . . . 6 ⊢ (𝑦 = 𝐵 → (( bday ‘𝐴) +no ( bday ‘𝑦)) = (( bday ‘𝐴) +no ( bday ‘𝐵)))
22	21	uneq1d 4114	. . . . 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 2846	. . . 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 7426	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
25	24	eleq1d 2846	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴 +s 𝑦) ∈ No ↔ (𝐴 +s 𝐵) ∈ No ))
26		breq1 5106	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 <s 𝑧 ↔ 𝐵 <s 𝑧))
27		oveq1 7425	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝐴) = (𝐵 +s 𝐴))
28	27	breq1d 5113	. . . . . 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 6883	. . . . . . 7 ⊢ (𝑧 = 𝐶 → ( bday ‘𝑧) = ( bday ‘𝐶))
33	32	oveq2d 7434	. . . . . 6 ⊢ (𝑧 = 𝐶 → (( bday ‘𝐴) +no ( bday ‘𝑧)) = (( bday ‘𝐴) +no ( bday ‘𝐶)))
34	33	uneq2d 4115	. . . . 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 2846	. . . 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 5107	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝐵 <s 𝑧 ↔ 𝐵 <s 𝐶))
37		oveq1 7425	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑧 +s 𝐴) = (𝐶 +s 𝐴))
38	37	breq2d 5115	. . . . . 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 3592	. 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 2145  ∀wral 3077   ∪ cun 3897   class class class wbr 5103  ‘cfv 6537  (class class class)co 7418   +no cnadd 8667   No csur 27990   <s clts 27991   bday cbday 27992   +_s cadds 28338

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6493  df-fv 6545  df-ov 7421

This theorem is used by:  addsproplem2  28349  addsproplem6  28353