Theorem negsval 28457

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

Assertion

Ref	Expression
negsval	⊢ (𝐴 ∈ No → ( -s‘𝐴) = (( -s “ (R‘𝐴)) |s ( -s “ (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 28453	. . 3 ⊢ -s = norec ((𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ (R‘𝑥)) |s (𝑛 “ (L‘𝑥)))))
2	1	norecov 28379	. 2 ⊢ (𝐴 ∈ No → ( -s‘𝐴) = (𝐴(𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ (R‘𝑥)) |s (𝑛 “ (L‘𝑥))))( -s ↾ ((L‘𝐴) ∪ (R‘𝐴)))))
3		elex 3475	. . 3 ⊢ (𝐴 ∈ No → 𝐴 ∈ V)
4		negsfn 28455	. . . . . 6 ⊢ -s Fn No
5		fnfun 6667	. . . . . 6 ⊢ ( -s Fn No → Fun -s)
6	4, 5	ax-mp 5	. . . . 5 ⊢ Fun -s
7		fvex 6927	. . . . . 6 ⊢ (L‘𝐴) ∈ V
8		fvex 6927	. . . . . 6 ⊢ (R‘𝐴) ∈ V
9	7, 8	unex 7791	. . . . 5 ⊢ ((L‘𝐴) ∪ (R‘𝐴)) ∈ V
10		resfunexg 7250	. . . . 5 ⊢ ((Fun -s ∧ ((L‘𝐴) ∪ (R‘𝐴)) ∈ V) → ( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) ∈ V)
11	6, 9, 10	mp2an 705	. . . 4 ⊢ ( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) ∈ V
12	11	a1i 11	. . 3 ⊢ (𝐴 ∈ No → ( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) ∈ V)
13		ovexd 7484	. . 3 ⊢ (𝐴 ∈ No → ((( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (R‘𝐴)) |s (( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (L‘𝐴))) ∈ V)
14		fveq2 6914	. . . . . 6 ⊢ (𝑥 = 𝐴 → (R‘𝑥) = (R‘𝐴))
15	14	imaeq2d 6068	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑛 “ (R‘𝑥)) = (𝑛 “ (R‘𝐴)))
16		fveq2 6914	. . . . . 6 ⊢ (𝑥 = 𝐴 → (L‘𝑥) = (L‘𝐴))
17	16	imaeq2d 6068	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑛 “ (L‘𝑥)) = (𝑛 “ (L‘𝐴)))
18	15, 17	oveq12d 7467	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑛 “ (R‘𝑥)) |s (𝑛 “ (L‘𝑥))) = ((𝑛 “ (R‘𝐴)) |s (𝑛 “ (L‘𝐴))))
19		imaeq1 6063	. . . . 5 ⊢ (𝑛 = ( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) → (𝑛 “ (R‘𝐴)) = (( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (R‘𝐴)))
20		imaeq1 6063	. . . . 5 ⊢ (𝑛 = ( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) → (𝑛 “ (L‘𝐴)) = (( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (L‘𝐴)))
21	19, 20	oveq12d 7467	. . . 4 ⊢ (𝑛 = ( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) → ((𝑛 “ (R‘𝐴)) |s (𝑛 “ (L‘𝐴))) = ((( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (R‘𝐴)) |s (( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (L‘𝐴))))
22		eqid 2763	. . . 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 7609	. . 3 ⊢ ((𝐴 ∈ V ∧ ( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) ∈ V ∧ ((( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (R‘𝐴)) |s (( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (L‘𝐴))) ∈ V) → (𝐴(𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ (R‘𝑥)) |s (𝑛 “ (L‘𝑥))))( -s ↾ ((L‘𝐴) ∪ (R‘𝐴)))) = ((( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (R‘𝐴)) |s (( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (L‘𝐴))))
24	3, 12, 13, 23	syl3anc 1399	. 2 ⊢ (𝐴 ∈ No → (𝐴(𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ (R‘𝑥)) |s (𝑛 “ (L‘𝑥))))( -s ↾ ((L‘𝐴) ∪ (R‘𝐴)))) = ((( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (R‘𝐴)) |s (( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (L‘𝐴))))
25		ssun2 4180	. . . . 5 ⊢ (R‘𝐴) ⊆ ((L‘𝐴) ∪ (R‘𝐴))
26		resima2 6073	. . . . 5 ⊢ ((R‘𝐴) ⊆ ((L‘𝐴) ∪ (R‘𝐴)) → (( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (R‘𝐴)) = ( -s “ (R‘𝐴)))
27	25, 26	ax-mp 5	. . . 4 ⊢ (( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (R‘𝐴)) = ( -s “ (R‘𝐴))
28		ssun1 4179	. . . . 5 ⊢ (L‘𝐴) ⊆ ((L‘𝐴) ∪ (R‘𝐴))
29		resima2 6073	. . . . 5 ⊢ ((L‘𝐴) ⊆ ((L‘𝐴) ∪ (R‘𝐴)) → (( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (L‘𝐴)) = ( -s “ (L‘𝐴)))
30	28, 29	ax-mp 5	. . . 4 ⊢ (( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (L‘𝐴)) = ( -s “ (L‘𝐴))
31	27, 30	oveq12i 7461	. . 3 ⊢ ((( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (R‘𝐴)) |s (( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (L‘𝐴))) = (( -s “ (R‘𝐴)) |s ( -s “ (L‘𝐴)))
32	31	a1i 11	. 2 ⊢ (𝐴 ∈ No → ((( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (R‘𝐴)) |s (( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (L‘𝐴))) = (( -s “ (R‘𝐴)) |s ( -s “ (L‘𝐴))))
33	2, 24, 32	3eqtrd 2802	1 ⊢ (𝐴 ∈ No → ( -s‘𝐴) = (( -s “ (R‘𝐴)) |s ( -s “ (L‘𝐴))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152   ↾ cres 5658   “ cima 5659  Fun wfun 6558   Fn wfn 6559  ‘cfv 6564  (class class class)co 7449   ∈ cmpo 7451  Nocsur 28043   |s ccuts 28191  Lcleft 28257  Rcright 28258   -_scnegs 28451

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-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-1o 8512  df-2o 8513  df-no 28046  df-lts 28047  df-bday 28048  df-slts 28190  df-cuts 28192  df-made 28259  df-old 28260  df-left 28262  df-right 28263  df-norec 28370  df-negs 28453

This theorem is used by:  neg0s  28458  neg1s  28459  negsproplem3  28462  negsid  28473  negsunif  28487  negbdaylem  28488