Theorem negsval 28417

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 28413	. . 3 ⊢ -s = norec ((𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ (R‘𝑥)) |s (𝑛 “ (L‘𝑥)))))
2	1	norecov 28339	. 2 ⊢ (𝐴 ∈ No → ( -s‘𝐴) = (𝐴(𝑥 ∈ V, 𝑛 ∈ V ↦ ((𝑛 “ (R‘𝑥)) |s (𝑛 “ (L‘𝑥))))( -s ↾ ((L‘𝐴) ∪ (R‘𝐴)))))
3		elex 3471	. . 3 ⊢ (𝐴 ∈ No → 𝐴 ∈ V)
4		negsfn 28415	. . . . . 6 ⊢ -s Fn No
5		fnfun 6640	. . . . . 6 ⊢ ( -s Fn No → Fun -s)
6	4, 5	ax-mp 5	. . . . 5 ⊢ Fun -s
7		fvex 6899	. . . . . 6 ⊢ (L‘𝐴) ∈ V
8		fvex 6899	. . . . . 6 ⊢ (R‘𝐴) ∈ V
9	7, 8	unex 7762	. . . . 5 ⊢ ((L‘𝐴) ∪ (R‘𝐴)) ∈ V
10		resfunexg 7222	. . . . 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 7456	. . 3 ⊢ (𝐴 ∈ No → ((( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (R‘𝐴)) |s (( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (L‘𝐴))) ∈ V)
14		fveq2 6886	. . . . . 6 ⊢ (𝑥 = 𝐴 → (R‘𝑥) = (R‘𝐴))
15	14	imaeq2d 6051	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑛 “ (R‘𝑥)) = (𝑛 “ (R‘𝐴)))
16		fveq2 6886	. . . . . 6 ⊢ (𝑥 = 𝐴 → (L‘𝑥) = (L‘𝐴))
17	16	imaeq2d 6051	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑛 “ (L‘𝑥)) = (𝑛 “ (L‘𝐴)))
18	15, 17	oveq12d 7439	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑛 “ (R‘𝑥)) |s (𝑛 “ (L‘𝑥))) = ((𝑛 “ (R‘𝐴)) |s (𝑛 “ (L‘𝐴))))
19		imaeq1 6046	. . . . 5 ⊢ (𝑛 = ( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) → (𝑛 “ (R‘𝐴)) = (( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (R‘𝐴)))
20		imaeq1 6046	. . . . 5 ⊢ (𝑛 = ( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) → (𝑛 “ (L‘𝐴)) = (( -s ↾ ((L‘𝐴) ∪ (R‘𝐴))) “ (L‘𝐴)))
21	19, 20	oveq12d 7439	. . . 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 2760	. . . 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 7580	. . 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 1398	. 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 4177	. . . . 5 ⊢ (R‘𝐴) ⊆ ((L‘𝐴) ∪ (R‘𝐴))
26		resima2 6056	. . . . 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 4176	. . . . 5 ⊢ (L‘𝐴) ⊆ ((L‘𝐴) ∪ (R‘𝐴))
29		resima2 6056	. . . . 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 7433	. . 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 2799	1 ⊢ (𝐴 ∈ No → ( -s‘𝐴) = (( -s “ (R‘𝐴)) |s ( -s “ (L‘𝐴))))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3895   ∪ cun 4149   ↾ cres 5652   “ cima 5653  Fun wfun 6533   Fn wfn 6534  ‘cfv 6539  (class class class)co 7421   ∈ cmpo 7423  Nocsur 28003   |s ccuts 28151  Lcleft 28217  Rcright 28218   -_scnegs 28411

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 7752

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 4585  df-pr 4587  df-tp 4589  df-op 4835  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 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-1o 8480  df-2o 8481  df-no 28006  df-lts 28007  df-bday 28008  df-slts 28150  df-cuts 28152  df-made 28219  df-old 28220  df-left 28222  df-right 28223  df-norec 28330  df-negs 28413

This theorem is used by:  neg0s  28418  neg1s  28419  negsproplem3  28422  negsid  28433  negsunif  28447  negbdaylem  28448