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Theorem adjvalval 31969

Description: Value of the value of the adjoint function. (Contributed by NM, 22-Feb-2006.) (Proof shortened by Mario Carneiro, 10-Sep-2015.) (New usage is discouraged.)

**Assertion**
Ref	Expression
adjvalval	⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝐴) = (℩𝑤 ∈ ℋ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝐴) = (𝑥 ·_ih 𝑤)))

Distinct variable groups: 𝑥,𝑤,𝐴 𝑥,𝑇,𝑤

**Proof of Theorem adjvalval**
Step	Hyp	Ref	Expression
1		adjcl 31964	. . 3 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝐴) ∈ ℋ)
2		eqcom 2747	. . . . . . 7 ⊢ (((𝑇‘𝑥) ·_ih 𝐴) = (𝑥 ·_ih 𝑤) ↔ (𝑥 ·_ih 𝑤) = ((𝑇‘𝑥) ·_ih 𝐴))
3		adj2 31966	. . . . . . . . . 10 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑥 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑇‘𝑥) ·_ih 𝐴) = (𝑥 ·_ih ((adj_ℎ‘𝑇)‘𝐴)))
4	3	3com23 1126	. . . . . . . . 9 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·_ih 𝐴) = (𝑥 ·_ih ((adj_ℎ‘𝑇)‘𝐴)))
5	4	3expa 1118	. . . . . . . 8 ⊢ (((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·_ih 𝐴) = (𝑥 ·_ih ((adj_ℎ‘𝑇)‘𝐴)))
6	5	eqeq2d 2751	. . . . . . 7 ⊢ (((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑥 ·_ih 𝑤) = ((𝑇‘𝑥) ·_ih 𝐴) ↔ (𝑥 ·_ih 𝑤) = (𝑥 ·_ih ((adj_ℎ‘𝑇)‘𝐴))))
7	2, 6	bitrid 283	. . . . . 6 ⊢ (((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → (((𝑇‘𝑥) ·_ih 𝐴) = (𝑥 ·_ih 𝑤) ↔ (𝑥 ·_ih 𝑤) = (𝑥 ·_ih ((adj_ℎ‘𝑇)‘𝐴))))
8	7	ralbidva 3182	. . . . 5 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) → (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝐴) = (𝑥 ·_ih 𝑤) ↔ ∀𝑥 ∈ ℋ (𝑥 ·_ih 𝑤) = (𝑥 ·_ih ((adj_ℎ‘𝑇)‘𝐴))))
9	8	adantr 480	. . . 4 ⊢ (((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝐴) = (𝑥 ·_ih 𝑤) ↔ ∀𝑥 ∈ ℋ (𝑥 ·_ih 𝑤) = (𝑥 ·_ih ((adj_ℎ‘𝑇)‘𝐴))))
10		simpr 484	. . . . 5 ⊢ (((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑤 ∈ ℋ)
11	1	adantr 480	. . . . 5 ⊢ (((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝐴) ∈ ℋ)
12		hial2eq2 31139	. . . . 5 ⊢ ((𝑤 ∈ ℋ ∧ ((adj_ℎ‘𝑇)‘𝐴) ∈ ℋ) → (∀𝑥 ∈ ℋ (𝑥 ·_ih 𝑤) = (𝑥 ·_ih ((adj_ℎ‘𝑇)‘𝐴)) ↔ 𝑤 = ((adj_ℎ‘𝑇)‘𝐴)))
13	10, 11, 12	syl2anc 583	. . . 4 ⊢ (((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (∀𝑥 ∈ ℋ (𝑥 ·_ih 𝑤) = (𝑥 ·_ih ((adj_ℎ‘𝑇)‘𝐴)) ↔ 𝑤 = ((adj_ℎ‘𝑇)‘𝐴)))
14	9, 13	bitrd 279	. . 3 ⊢ (((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → (∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝐴) = (𝑥 ·_ih 𝑤) ↔ 𝑤 = ((adj_ℎ‘𝑇)‘𝐴)))
15	1, 14	riota5 7434	. 2 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) → (℩𝑤 ∈ ℋ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝐴) = (𝑥 ·_ih 𝑤)) = ((adj_ℎ‘𝑇)‘𝐴))
16	15	eqcomd 2746	1 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝐴) = (℩𝑤 ∈ ℋ ∀𝑥 ∈ ℋ ((𝑇‘𝑥) ·_ih 𝐴) = (𝑥 ·_ih 𝑤)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 dom cdm 5700 ‘cfv 6573 ℩crio 7403 (class class class)co 7448 ℋchba 30951 ·_ih csp 30954 adj_ℎcado 30987

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-hilex 31031 ax-hfvadd 31032 ax-hvcom 31033 ax-hvass 31034 ax-hv0cl 31035 ax-hvaddid 31036 ax-hfvmul 31037 ax-hvmulid 31038 ax-hvdistr2 31041 ax-hvmul0 31042 ax-hfi 31111 ax-his1 31114 ax-his2 31115 ax-his3 31116 ax-his4 31117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-2 12356 df-cj 15148 df-re 15149 df-im 15150 df-hvsub 31003 df-adjh 31881

This theorem is referenced by: nmopadjlei 32120

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