Theorem negsproplem4 28417

Description: Lemma for surreal negation. Show the second half of the inductive hypothesis when 𝐴 is simpler than 𝐵. (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)))))
negsproplem4.1	⊢ (𝜑 → 𝐴 ∈ No)
negsproplem4.2	⊢ (𝜑 → 𝐵 ∈ No)
negsproplem4.3	⊢ (𝜑 → 𝐴 <s 𝐵)
negsproplem4.4	⊢ (𝜑 → (bday‘𝐴) ∈ (bday‘𝐵))

Assertion

Ref	Expression
negsproplem4	⊢ (𝜑 → ( -s‘𝐵) <s ( -s‘𝐴))

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

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem negsproplem4

Step	Hyp	Ref	Expression
1		negsproplem.1	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
2		uncom 4157	. . . . . . . 8 ⊢ ((bday‘𝐴) ∪ (bday‘𝐵)) = ((bday‘𝐵) ∪ (bday‘𝐴))
3	2	eleq2i 2852	. . . . . . 7 ⊢ (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) ↔ ((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐵) ∪ (bday‘𝐴)))
4	3	imbi1i 352	. . . . . 6 ⊢ ((((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) ↔ (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐵) ∪ (bday‘𝐴)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
5	4	2ralbii 3137	. . . . 5 ⊢ (∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐴) ∪ (bday‘𝐵)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐵) ∪ (bday‘𝐴)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
6	1, 5	sylib 221	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ No ∀𝑦 ∈ No (((bday‘𝑥) ∪ (bday‘𝑦)) ∈ ((bday‘𝐵) ∪ (bday‘𝐴)) → (( -s‘𝑥) ∈ No ∧ (𝑥 <s 𝑦 → ( -s‘𝑦) <s ( -s‘𝑥)))))
7		negsproplem4.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ No)
8	6, 7	negsproplem3 28416	. . 3 ⊢ (𝜑 → (( -s‘𝐵) ∈ No ∧ ( -s “ (R‘𝐵)) <<s {( -s‘𝐵)} ∧ {( -s‘𝐵)} <<s ( -s “ (L‘𝐵))))
9	8	simp3d 1162	. 2 ⊢ (𝜑 → {( -s‘𝐵)} <<s ( -s “ (L‘𝐵)))
10		fvex 6898	. . . 4 ⊢ ( -s‘𝐵) ∈ V
11	10	snid 4622	. . 3 ⊢ ( -s‘𝐵) ∈ {( -s‘𝐵)}
12	11	a1i 11	. 2 ⊢ (𝜑 → ( -s‘𝐵) ∈ {( -s‘𝐵)})
13		negsfn 28409	. . 3 ⊢ -s Fn No
14		leftssno 28259	. . 3 ⊢ (L‘𝐵) ⊆ No
15		negsproplem4.4	. . . . 5 ⊢ (𝜑 → (bday‘𝐴) ∈ (bday‘𝐵))
16		bdayon 28138	. . . . . 6 ⊢ (bday‘𝐵) ∈ On
17		negsproplem4.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ No)
18		oldbday 28287	. . . . . 6 ⊢ (((bday‘𝐵) ∈ On ∧ 𝐴 ∈ No) → (𝐴 ∈ (O‘(bday‘𝐵)) ↔ (bday‘𝐴) ∈ (bday‘𝐵)))
19	16, 17, 18	sylancr 599	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ (O‘(bday‘𝐵)) ↔ (bday‘𝐴) ∈ (bday‘𝐵)))
20	15, 19	mpbird 260	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (O‘(bday‘𝐵)))
21		negsproplem4.3	. . . 4 ⊢ (𝜑 → 𝐴 <s 𝐵)
22		elleft 28237	. . . 4 ⊢ (𝐴 ∈ (L‘𝐵) ↔ (𝐴 ∈ (O‘(bday‘𝐵)) ∧ 𝐴 <s 𝐵))
23	20, 21, 22	sylanbrc 595	. . 3 ⊢ (𝜑 → 𝐴 ∈ (L‘𝐵))
24		fnfvima 7239	. . 3 ⊢ (( -s Fn No ∧ (L‘𝐵) ⊆ No ∧ 𝐴 ∈ (L‘𝐵)) → ( -s‘𝐴) ∈ ( -s “ (L‘𝐵)))
25	13, 14, 23, 24	mp3an12i 1494	. 2 ⊢ (𝜑 → ( -s‘𝐴) ∈ ( -s “ (L‘𝐵)))
26	9, 12, 25	sltssepcd 28158	1 ⊢ (𝜑 → ( -s‘𝐵) <s ( -s‘𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   ∪ cun 4149  {csn 4583   class class class wbr 5102   “ cima 5653  Oncon0 6362   Fn wfn 6533  ‘cfv 6538  Nocsur 27997   <s clts 27998  bdaycbday 27999   <<s cslts 28143  Ocold 28209  Lcleft 28211  Rcright 28212   -_scnegs 28405

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-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  df-no 28000  df-lts 28001  df-bday 28002  df-slts 28144  df-cuts 28146  df-0s 28193  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec 28324  df-negs 28407

This theorem is used by:  negsproplem7  28420