Theorem addsproplem1 27977

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 8340 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 1129	. 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 6842	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ( bday ‘𝑥) = ( bday ‘𝐴))
8	7	oveq1d 7383	. . . . . 6 ⊢ (𝑥 = 𝐴 → (( bday ‘𝑥) +no ( bday ‘𝑦)) = (( bday ‘𝐴) +no ( bday ‘𝑦)))
9	7	oveq1d 7383	. . . . . 6 ⊢ (𝑥 = 𝐴 → (( bday ‘𝑥) +no ( bday ‘𝑧)) = (( bday ‘𝐴) +no ( bday ‘𝑧)))
10	8, 9	uneq12d 4123	. . . . 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 2822	. . . 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 7375	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
13	12	eleq1d 2822	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 +s 𝑦) ∈ No ↔ (𝐴 +s 𝑦) ∈ No ))
14		oveq2 7376	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 +s 𝑥) = (𝑦 +s 𝐴))
15		oveq2 7376	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑧 +s 𝑥) = (𝑧 +s 𝐴))
16	14, 15	breq12d 5113	. . . . . 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 633	. . . 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 6842	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ( bday ‘𝑦) = ( bday ‘𝐵))
21	20	oveq2d 7384	. . . . . 6 ⊢ (𝑦 = 𝐵 → (( bday ‘𝐴) +no ( bday ‘𝑦)) = (( bday ‘𝐴) +no ( bday ‘𝐵)))
22	21	uneq1d 4121	. . . . 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 2822	. . . 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 7376	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
25	24	eleq1d 2822	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴 +s 𝑦) ∈ No ↔ (𝐴 +s 𝐵) ∈ No ))
26		breq1 5103	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 <s 𝑧 ↔ 𝐵 <s 𝑧))
27		oveq1 7375	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝐴) = (𝐵 +s 𝐴))
28	27	breq1d 5110	. . . . . 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 633	. . . 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 6842	. . . . . . 7 ⊢ (𝑧 = 𝐶 → ( bday ‘𝑧) = ( bday ‘𝐶))
33	32	oveq2d 7384	. . . . . 6 ⊢ (𝑧 = 𝐶 → (( bday ‘𝐴) +no ( bday ‘𝑧)) = (( bday ‘𝐴) +no ( bday ‘𝐶)))
34	33	uneq2d 4122	. . . . 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 2822	. . . 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 5104	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝐵 <s 𝑧 ↔ 𝐵 <s 𝐶))
37		oveq1 7375	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑧 +s 𝐴) = (𝐶 +s 𝐴))
38	37	breq2d 5112	. . . . . 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 631	. . . 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 3594	. 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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∪ cun 3901   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368   +no cnadd 8603   No csur 27619   <s clts 27620   bday cbday 27621   +_s cadds 27967

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-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371

This theorem is referenced by:  addsproplem2  27978  addsproplem6  27982