Theorem negsval 28017

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 28013	. . 3 ⊢ -us = norec ((𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥)))))
2	1	norecov 27939	. 2 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) = (𝐴(𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))))( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴)))))
3		elex 3450	. . 3 ⊢ (𝐴 ∈ No → 𝐴 ∈ V)
4		negsfn 28015	. . . . . 6 ⊢ -us Fn No
5		fnfun 6598	. . . . . 6 ⊢ ( -us Fn No → Fun -us )
6	4, 5	ax-mp 5	. . . . 5 ⊢ Fun -us
7		fvex 6853	. . . . . 6 ⊢ ( L ‘𝐴) ∈ V
8		fvex 6853	. . . . . 6 ⊢ ( R ‘𝐴) ∈ V
9	7, 8	unex 7698	. . . . 5 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) ∈ V
10		resfunexg 7170	. . . . 5 ⊢ ((Fun -us ∧ (( L ‘𝐴) ∪ ( R ‘𝐴)) ∈ V) → ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) ∈ V)
11	6, 9, 10	mp2an 693	. . . 4 ⊢ ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) ∈ V
12	11	a1i 11	. . 3 ⊢ (𝐴 ∈ No → ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) ∈ V)
13		ovexd 7402	. . 3 ⊢ (𝐴 ∈ No → ((( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)) |s (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴))) ∈ V)
14		fveq2 6840	. . . . . 6 ⊢ (𝑥 = 𝐴 → ( R ‘𝑥) = ( R ‘𝐴))
15	14	imaeq2d 6025	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑛 “ ( R ‘𝑥)) = (𝑛 “ ( R ‘𝐴)))
16		fveq2 6840	. . . . . 6 ⊢ (𝑥 = 𝐴 → ( L ‘𝑥) = ( L ‘𝐴))
17	16	imaeq2d 6025	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑛 “ ( L ‘𝑥)) = (𝑛 “ ( L ‘𝐴)))
18	15, 17	oveq12d 7385	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑛 “ ( R ‘𝑥)) |s (𝑛 “ ( L ‘𝑥))) = ((𝑛 “ ( R ‘𝐴)) |s (𝑛 “ ( L ‘𝐴))))
19		imaeq1 6020	. . . . 5 ⊢ (𝑛 = ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) → (𝑛 “ ( R ‘𝐴)) = (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( R ‘𝐴)))
20		imaeq1 6020	. . . . 5 ⊢ (𝑛 = ( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) → (𝑛 “ ( L ‘𝐴)) = (( -us ↾ (( L ‘𝐴) ∪ ( R ‘𝐴))) “ ( L ‘𝐴)))
21	19, 20	oveq12d 7385	. . . 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 2736	. . . 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 7526	. . 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 1374	. 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 4119	. . . . 5 ⊢ ( R ‘𝐴) ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴))
26		resima2 5981	. . . . 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 4118	. . . . 5 ⊢ ( L ‘𝐴) ⊆ (( L ‘𝐴) ∪ ( R ‘𝐴))
29		resima2 5981	. . . . 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 7379	. . 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 2775	1 ⊢ (𝐴 ∈ No → ( -us ‘𝐴) = (( -us “ ( R ‘𝐴)) |s ( -us “ ( L ‘𝐴))))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∪ cun 3887   ⊆ wss 3889   ↾ cres 5633   “ cima 5634  Fun wfun 6492   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369   No csur 27603   |s ccuts 27751   L cleft 27817   R cright 27818   -u_s cnegs 28011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608  df-slts 27750  df-cuts 27752  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-norec 27930  df-negs 28013

This theorem is referenced by:  neg0s  28018  neg1s  28019  negsproplem3  28022  negsid  28033  negsunif  28047  negbdaylem  28048