Theorem addsproplem1 27913

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 8318 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 1128	. 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 6822	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ( bday ‘𝑥) = ( bday ‘𝐴))
8	7	oveq1d 7361	. . . . . 6 ⊢ (𝑥 = 𝐴 → (( bday ‘𝑥) +no ( bday ‘𝑦)) = (( bday ‘𝐴) +no ( bday ‘𝑦)))
9	7	oveq1d 7361	. . . . . 6 ⊢ (𝑥 = 𝐴 → (( bday ‘𝑥) +no ( bday ‘𝑧)) = (( bday ‘𝐴) +no ( bday ‘𝑧)))
10	8, 9	uneq12d 4119	. . . . 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 2816	. . . 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 7353	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
13	12	eleq1d 2816	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 +s 𝑦) ∈ No ↔ (𝐴 +s 𝑦) ∈ No ))
14		oveq2 7354	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 +s 𝑥) = (𝑦 +s 𝐴))
15		oveq2 7354	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑧 +s 𝑥) = (𝑧 +s 𝐴))
16	14, 15	breq12d 5104	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑦 +s 𝑥) <s (𝑧 +s 𝑥) ↔ (𝑦 +s 𝐴) <s (𝑧 +s 𝐴)))
17	16	imbi2d 340	. . . . 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 632	. . . 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 344	. . 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 6822	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ( bday ‘𝑦) = ( bday ‘𝐵))
21	20	oveq2d 7362	. . . . . 6 ⊢ (𝑦 = 𝐵 → (( bday ‘𝐴) +no ( bday ‘𝑦)) = (( bday ‘𝐴) +no ( bday ‘𝐵)))
22	21	uneq1d 4117	. . . . 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 2816	. . . 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 7354	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
25	24	eleq1d 2816	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴 +s 𝑦) ∈ No ↔ (𝐴 +s 𝐵) ∈ No ))
26		breq1 5094	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 <s 𝑧 ↔ 𝐵 <s 𝑧))
27		oveq1 7353	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝐴) = (𝐵 +s 𝐴))
28	27	breq1d 5101	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝑦 +s 𝐴) <s (𝑧 +s 𝐴) ↔ (𝐵 +s 𝐴) <s (𝑧 +s 𝐴)))
29	26, 28	imbi12d 344	. . . . 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 632	. . . 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 344	. . 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 6822	. . . . . . 7 ⊢ (𝑧 = 𝐶 → ( bday ‘𝑧) = ( bday ‘𝐶))
33	32	oveq2d 7362	. . . . . 6 ⊢ (𝑧 = 𝐶 → (( bday ‘𝐴) +no ( bday ‘𝑧)) = (( bday ‘𝐴) +no ( bday ‘𝐶)))
34	33	uneq2d 4118	. . . . 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 2816	. . . 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 5095	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝐵 <s 𝑧 ↔ 𝐵 <s 𝐶))
37		oveq1 7353	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑧 +s 𝐴) = (𝐶 +s 𝐴))
38	37	breq2d 5103	. . . . . 6 ⊢ (𝑧 = 𝐶 → ((𝐵 +s 𝐴) <s (𝑧 +s 𝐴) ↔ (𝐵 +s 𝐴) <s (𝐶 +s 𝐴)))
39	36, 38	imbi12d 344	. . . . 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 630	. . . 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 344	. . 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 3593	. 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 66	1 ⊢ (𝜑 → ((𝐴 +s 𝐵) ∈ No ∧ (𝐵 <s 𝐶 → (𝐵 +s 𝐴) <s (𝐶 +s 𝐴))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∀wral 3047   ∪ cun 3900   class class class wbr 5091  ‘cfv 6481  (class class class)co 7346   +no cnadd 8580   No csur 27579   <s cslt 27580   bday cbday 27581   +_s cadds 27903

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-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-iota 6437  df-fv 6489  df-ov 7349

This theorem is referenced by:  addsproplem2  27914  addsproplem6  27918