Theorem negsval 28032

Description: The value of the surreal negation function. (Contributed by Scott Fenton, 20-Aug-2024.)

Assertion

Ref	Expression
negsval	⊢ (𝐴 ∈ No → ( -us ‘𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))

Proof of Theorem negsval

Dummy variables 𝑥 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-negs 28028	. . 3 ⊢ -us = norec ((𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥)))))
2	1	norecov 27958	. 2 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) = (𝐴(𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))))( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴)))))
3		elex 3493	. . 3 ⊢ (𝐴 ∈ No → 𝐴 ∈ V)
4		negsfn 28030	. . . . . 6 ⊢ -us Fn No
5		fnfun 6662	. . . . . 6 ⊢ ( -us Fn No → Fun -us )
6	4, 5	ax-mp 5	. . . . 5 ⊢ Fun -us
7		fvex 6916	. . . . . 6 ⊢ ( L ‘𝐴) ∈ V
8		fvex 6916	. . . . . 6 ⊢ ( R ‘𝐴) ∈ V
9	7, 8	unex 7757	. . . . 5 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) ∈ V
10		resfunexg 7234	. . . . 5 ⊢ ((Fun -us ∧ (( L ‘𝐴) ∪ ( R ‘𝐴)) ∈ V) → ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) ∈ V)
11	6, 9, 10	mp2an 690	. . . 4 ⊢ ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) ∈ V
12	11	a1i 11	. . 3 ⊢ (𝐴 ∈ No → ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) ∈ V)
13		ovexd 7462	. . 3 ⊢ (𝐴 ∈ No → ((( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) |s (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴))) ∈ V)
14		fveq2 6903	. . . . . 6 ⊢ (𝑥 = 𝐴 → ( R ‘𝑥) = ( R ‘𝐴))
15	14	imaeq2d 6072	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑛 “ ( R ‘𝑥)) = (𝑛 “ ( R ‘𝐴)))
16		fveq2 6903	. . . . . 6 ⊢ (𝑥 = 𝐴 → ( L ‘𝑥) = ( L ‘𝐴))
17	16	imaeq2d 6072	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑛 “ ( L ‘𝑥)) = (𝑛 “ ( L ‘𝐴)))
18	15, 17	oveq12d 7445	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))) = ((𝑛 “ ( R ‘𝐴)) |s (𝑛 “ ( L ‘𝐴))))
19		imaeq1 6067	. . . . 5 ⊢ (𝑛 = ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) → (𝑛 “ ( R ‘𝐴)) = (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)))
20		imaeq1 6067	. . . . 5 ⊢ (𝑛 = ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) → (𝑛 “ ( L ‘𝐴)) = (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴)))
21	19, 20	oveq12d 7445	. . . 4 ⊢ (𝑛 = ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) → ((𝑛 “ ( R ‘𝐴)) |s (𝑛 “ ( L ‘𝐴))) = ((( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) |s (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴))))
22		eqid 2729	. . . 4 ⊢ (𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥)))) = (𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))))
23	18, 21, 22	ovmpog 7587	. . 3 ⊢ ((𝐴 ∈ V ∧ ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) ∈ V ∧ ((( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) |s (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴))) ∈ V) → (𝐴(𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))))( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴)))) = ((( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) |s (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴))))
24	3, 12, 13, 23	syl3anc 1368	. 2 ⊢ (𝐴 ∈ No → (𝐴(𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))))( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴)))) = ((( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) |s (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴))))
25		ssun2 4182	. . . . 5 ⊢ ( R ‘𝐴) ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴))
26		resima2 6028	. . . . 5 ⊢ (( R ‘𝐴) ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴)) → (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) = ( -us “ ( R ‘𝐴)))
27	25, 26	ax-mp 5	. . . 4 ⊢ (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) = ( -us “ ( R ‘𝐴))
28		ssun1 4181	. . . . 5 ⊢ ( L ‘𝐴) ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴))
29		resima2 6028	. . . . 5 ⊢ (( L ‘𝐴) ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴)) → (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴)) = ( -us “ ( L ‘𝐴)))
30	28, 29	ax-mp 5	. . . 4 ⊢ (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴)) = ( -us “ ( L ‘𝐴))
31	27, 30	oveq12i 7439	. . 3 ⊢ ((( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) |s (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴))) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴)))
32	31	a1i 11	. 2 ⊢ (𝐴 ∈ No → ((( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) |s (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴))) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))
33	2, 24, 32	3eqtrd 2773	1 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2100  Vcvv 3472   ∪ cun 3956   ⊆ wss 3958   ↾ cres 5686   “ cima 5687  Fun wfun 6550   Fn wfn 6551  ‘cfv 6556  (class class class)co 7427   ∈ cmpo 7429   No csur 27666   |s cscut 27809   L cleft 27866   R cright 27867   -u_s cnegs 28026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-rep 5291  ax-sep 5305  ax-nul 5312  ax-pow 5371  ax-pr 5435  ax-un 7748

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-ral 3055  df-rex 3064  df-rmo 3373  df-reu 3374  df-rab 3429  df-v 3474  df-sbc 3788  df-csb 3904  df-dif 3961  df-un 3963  df-in 3965  df-ss 3975  df-pss 3978  df-nul 4334  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-tp 4639  df-op 4641  df-uni 4917  df-int 4958  df-iun 5006  df-br 5155  df-opab 5217  df-mpt 5238  df-tr 5272  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5639  df-se 5640  df-we 5641  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6315  df-ord 6381  df-on 6382  df-suc 6384  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-f1 6561  df-fo 6562  df-f1o 6563  df-fv 6564  df-riota 7383  df-ov 7430  df-oprab 7431  df-mpo 7432  df-2nd 8009  df-frecs 8300  df-wrecs 8331  df-recs 8405  df-1o 8500  df-2o 8501  df-no 27669  df-slt 27670  df-bday 27671  df-sslt 27808  df-scut 27810  df-made 27868  df-old 27869  df-left 27871  df-right 27872  df-norec 27949  df-negs 28028

This theorem is referenced by:  negs0s  28033  negs1s  28034  negsproplem3  28037  negsid  28048  negsunif  28062  negsbdaylem  28063