Theorem negsval 27987

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 27983	. . 3 ⊢ -us = norec ((𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥)))))
2	1	norecov 27910	. 2 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) = (𝐴(𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))))( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴)))))
3		elex 3458	. . 3 ⊢ (𝐴 ∈ No → 𝐴 ∈ V)
4		negsfn 27985	. . . . . 6 ⊢ -us Fn No
5		fnfun 6589	. . . . . 6 ⊢ ( -us Fn No → Fun -us )
6	4, 5	ax-mp 5	. . . . 5 ⊢ Fun -us
7		fvex 6844	. . . . . 6 ⊢ ( L ‘𝐴) ∈ V
8		fvex 6844	. . . . . 6 ⊢ ( R ‘𝐴) ∈ V
9	7, 8	unex 7686	. . . . 5 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) ∈ V
10		resfunexg 7158	. . . . 5 ⊢ ((Fun -us ∧ (( L ‘𝐴) ∪ ( R ‘𝐴)) ∈ V) → ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) ∈ V)
11	6, 9, 10	mp2an 692	. . . 4 ⊢ ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) ∈ V
12	11	a1i 11	. . 3 ⊢ (𝐴 ∈ No → ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) ∈ V)
13		ovexd 7390	. . 3 ⊢ (𝐴 ∈ No → ((( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) |s (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴))) ∈ V)
14		fveq2 6831	. . . . . 6 ⊢ (𝑥 = 𝐴 → ( R ‘𝑥) = ( R ‘𝐴))
15	14	imaeq2d 6016	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑛 “ ( R ‘𝑥)) = (𝑛 “ ( R ‘𝐴)))
16		fveq2 6831	. . . . . 6 ⊢ (𝑥 = 𝐴 → ( L ‘𝑥) = ( L ‘𝐴))
17	16	imaeq2d 6016	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑛 “ ( L ‘𝑥)) = (𝑛 “ ( L ‘𝐴)))
18	15, 17	oveq12d 7373	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))) = ((𝑛 “ ( R ‘𝐴)) |s (𝑛 “ ( L ‘𝐴))))
19		imaeq1 6011	. . . . 5 ⊢ (𝑛 = ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) → (𝑛 “ ( R ‘𝐴)) = (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)))
20		imaeq1 6011	. . . . 5 ⊢ (𝑛 = ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) → (𝑛 “ ( L ‘𝐴)) = (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴)))
21	19, 20	oveq12d 7373	. . . 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 2733	. . . 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 7514	. . 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 1373	. 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 4128	. . . . 5 ⊢ ( R ‘𝐴) ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴))
26		resima2 5972	. . . . 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 4127	. . . . 5 ⊢ ( L ‘𝐴) ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴))
29		resima2 5972	. . . . 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 7367	. . 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 2772	1 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3437   ∪ cun 3896   ⊆ wss 3898   ↾ cres 5623   “ cima 5624  Fun wfun 6483   Fn wfn 6484  ‘cfv 6489  (class class class)co 7355   ∈ cmpo 7357   No csur 27598   |s cscut 27742   L cleft 27806   R cright 27807   -u_s cnegs 27981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-1o 8394  df-2o 8395  df-no 27601  df-slt 27602  df-bday 27603  df-sslt 27741  df-scut 27743  df-made 27808  df-old 27809  df-left 27811  df-right 27812  df-norec 27901  df-negs 27983

This theorem is referenced by:  negs0s  27988  negs1s  27989  negsproplem3  27992  negsid  28003  negsunif  28017  negsbdaylem  28018