Theorem addsproplem1 28364

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 8414 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 6890	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (bday‘𝑥) = (bday‘𝐴))
8	7	oveq1d 7439	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((bday‘𝑥) +no (bday‘𝑦)) = ((bday‘𝐴) +no (bday‘𝑦)))
9	7	oveq1d 7439	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((bday‘𝑥) +no (bday‘𝑧)) = ((bday‘𝐴) +no (bday‘𝑧)))
10	8, 9	uneq12d 4167	. . . . 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 2845	. . . 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 7431	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 +s 𝑦) = (𝐴 +s 𝑦))
13	12	eleq1d 2845	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥 +s 𝑦) ∈ No ↔ (𝐴 +s 𝑦) ∈ No))
14		oveq2 7432	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑦 +s 𝑥) = (𝑦 +s 𝐴))
15		oveq2 7432	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑧 +s 𝑥) = (𝑧 +s 𝐴))
16	14, 15	breq12d 5115	. . . . . 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 6890	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (bday‘𝑦) = (bday‘𝐵))
21	20	oveq2d 7440	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((bday‘𝐴) +no (bday‘𝑦)) = ((bday‘𝐴) +no (bday‘𝐵)))
22	21	uneq1d 4165	. . . . 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 2845	. . . 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 7432	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
25	24	eleq1d 2845	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐴 +s 𝑦) ∈ No ↔ (𝐴 +s 𝐵) ∈ No))
26		breq1 5105	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 <s 𝑧 ↔ 𝐵 <s 𝑧))
27		oveq1 7431	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝑦 +s 𝐴) = (𝐵 +s 𝐴))
28	27	breq1d 5112	. . . . . 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 6890	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (bday‘𝑧) = (bday‘𝐶))
33	32	oveq2d 7440	. . . . . 6 ⊢ (𝑧 = 𝐶 → ((bday‘𝐴) +no (bday‘𝑧)) = ((bday‘𝐴) +no (bday‘𝐶)))
34	33	uneq2d 4166	. . . . 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 2845	. . . 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 5106	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝐵 <s 𝑧 ↔ 𝐵 <s 𝐶))
37		oveq1 7431	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝑧 +s 𝐴) = (𝐶 +s 𝐴))
38	37	breq2d 5114	. . . . . 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 3591	. 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 1569   ∈ wcel 2144  ∀wral 3076   ∪ cun 4148   class class class wbr 5102  ‘cfv 6543  (class class class)co 7424   +no cnadd 8681  Nocsur 28006   <s clts 28007  bdaycbday 28008   +_s cadds 28354

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6497  df-fv 6551  df-ov 7427

This theorem is used by:  addsproplem2  28365  addsproplem6  28369