Theorem negsproplem1 28441

Description: Lemma for surreal negation. We prove a pair of properties of surreal negation simultaneously. First, we instantiate some quantifiers. (Contributed by Scott Fenton, 2-Feb-2025.)

Hypotheses

Ref	Expression
negsproplem.1	⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
negsproplem1.1	⊢ (𝜑 → 𝑋 ∈ No)
negsproplem1.2	⊢ (𝜑 → 𝑌 ∈ No)
negsproplem1.3	⊢ (𝜑 → ((bday‘𝑋) ∪ (bday‘𝑌)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)))

Assertion

Ref	Expression
negsproplem1	⊢ (𝜑 → (( -s‘𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -s‘𝑌) <s ( -s‘𝑋))))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑋,𝑦   𝑦,𝑌

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝑌(𝑥)

Proof of Theorem negsproplem1

Step	Hyp	Ref	Expression
1		negsproplem1.1	. . 3 ⊢ (𝜑 → 𝑋 ∈ No)
2		negsproplem1.2	. . 3 ⊢ (𝜑 → 𝑌 ∈ No)
3	1, 2	jca 521	. 2 ⊢ (𝜑 → (𝑋 ∈ No ∧ 𝑌 ∈ No))
4		negsproplem.1	. 2 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
5		negsproplem1.3	. 2 ⊢ (𝜑 → ((bday‘𝑋) ∪ (bday‘𝑌)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)))
6		fveq2 6905	. . . . . 6 ⊢ (𝑥 = 𝑋 → (bday‘𝑥) = (bday‘𝑋))
7	6	uneq1d 4169	. . . . 5 ⊢ (𝑥 = 𝑋 → ((bday‘𝑥) ∪ (bday‘𝑦)) = ((bday‘𝑋) ∪ (bday‘𝑦)))
8	7	eleq1d 2849	. . . 4 ⊢ (𝑥 = 𝑋 → (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) ↔ ((bday‘𝑋) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵))))
9		fveq2 6905	. . . . . 6 ⊢ (𝑥 = 𝑋 → ( -s‘𝑥) = ( -s‘𝑋))
10	9	eleq1d 2849	. . . . 5 ⊢ (𝑥 = 𝑋 → (( -s‘𝑥) ∈ No ↔ ( -s‘𝑋) ∈ No))
11		breq1 5109	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 <s 𝑦 ↔ 𝑋 <s 𝑦))
12	9	breq2d 5118	. . . . . 6 ⊢ (𝑥 = 𝑋 → (( -s‘𝑦) <s ( -s‘𝑥) ↔ ( -s‘𝑦) <s ( -s‘𝑋)))
13	11, 12	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)) ↔ (𝑋 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑋))))
14	10, 13	anbi12d 644	. . . 4 ⊢ (𝑥 = 𝑋 → ((( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥))) ↔ (( -s‘𝑋) ∈ No ∧ (𝑋 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑋)))))
15	8, 14	imbi12d 347	. . 3 ⊢ (𝑥 = 𝑋 → ((((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) ↔ (((bday‘𝑋) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑋) ∈ No ∧ (𝑋 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑋))))))
16		fveq2 6905	. . . . . 6 ⊢ (𝑦 = 𝑌 → (bday‘𝑦) = (bday‘𝑌))
17	16	uneq2d 4170	. . . . 5 ⊢ (𝑦 = 𝑌 → ((bday‘𝑋) ∪ (bday‘𝑦)) = ((bday‘𝑋) ∪ (bday‘𝑌)))
18	17	eleq1d 2849	. . . 4 ⊢ (𝑦 = 𝑌 → (((bday‘𝑋) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) ↔ ((bday‘𝑋) ∪ (bday‘𝑌)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵))))
19		breq2 5110	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 <s 𝑦 ↔ 𝑋 <s 𝑌))
20		fveq2 6905	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ( -s‘𝑦) = ( -s‘𝑌))
21	20	breq1d 5116	. . . . . 6 ⊢ (𝑦 = 𝑌 → (( -s‘𝑦) <s ( -s‘𝑋) ↔ ( -s‘𝑌) <s ( -s‘𝑋)))
22	19, 21	imbi12d 347	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑋)) ↔ (𝑋 <s 𝑌 → ( -s‘𝑌) <s ( -s‘𝑋))))
23	22	anbi2d 642	. . . 4 ⊢ (𝑦 = 𝑌 → ((( -s‘𝑋) ∈ No ∧ (𝑋 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑋))) ↔ (( -s‘𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -s‘𝑌) <s ( -s‘𝑋)))))
24	18, 23	imbi12d 347	. . 3 ⊢ (𝑦 = 𝑌 → ((((bday‘𝑋) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑋) ∈ No ∧ (𝑋 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑋)))) ↔ (((bday‘𝑋) ∪ (bday‘𝑌)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -s‘𝑌) <s ( -s‘𝑋))))))
25	15, 24	rspc2v 3590	. 2 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No) → (∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) → (((bday‘𝑋) ∪ (bday‘𝑌)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -s‘𝑌) <s ( -s‘𝑋))))))
26	3, 4, 5, 25	syl3c 67	1 ⊢ (𝜑 → (( -s‘𝑋) ∈ No ∧ (𝑋 <s 𝑌 → ( -s‘𝑌) <s ( -s‘𝑋))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ∪ cun 4152   class class class wbr 5106  ‘cfv 6555  Nocsur 28024   <s clts 28025  bdaycbday 28026   -_scnegs 28432

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 4839  df-uni 4871  df-br 5107  df-iota 6509  df-fv 6563

This theorem is used by:  negsproplem2  28442  negsproplem6  28446