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Theorem adjvalval 29368

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 29363	. . 3 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝐴) ∈ ℋ)
2		eqcom 2785	. . . . . . 7 ⊢ (((𝑇‘𝑥) ·_ih 𝐴) = (𝑥 ·_ih 𝑤) ↔ (𝑥 ·_ih 𝑤) = ((𝑇‘𝑥) ·_ih 𝐴))
3		adj2 29365	. . . . . . . . . 10 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝑥 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑇‘𝑥) ·_ih 𝐴) = (𝑥 ·_ih ((adj_ℎ‘𝑇)‘𝐴)))
4	3	3com23 1117	. . . . . . . . 9 ⊢ ((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·_ih 𝐴) = (𝑥 ·_ih ((adj_ℎ‘𝑇)‘𝐴)))
5	4	3expa 1108	. . . . . . . 8 ⊢ (((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑥 ∈ ℋ) → ((𝑇‘𝑥) ·_ih 𝐴) = (𝑥 ·_ih ((adj_ℎ‘𝑇)‘𝐴)))
6	5	eqeq2d 2788	. . . . . . 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	syl5bb 275	. . . . . 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 3167	. . . . 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 474	. . . 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 479	. . . . 5 ⊢ (((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → 𝑤 ∈ ℋ)
11	1	adantr 474	. . . . 5 ⊢ (((𝑇 ∈ dom adj_ℎ ∧ 𝐴 ∈ ℋ) ∧ 𝑤 ∈ ℋ) → ((adj_ℎ‘𝑇)‘𝐴) ∈ ℋ)
12		hial2eq2 28536	. . . . 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 579	. . . 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 271	. . 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 6909	. 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 2784	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 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 dom cdm 5355 ‘cfv 6135 ℩crio 6882 (class class class)co 6922 ℋchba 28348 ·_ih csp 28351 adj_ℎcado 28384

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-hilex 28428 ax-hfvadd 28429 ax-hvcom 28430 ax-hvass 28431 ax-hv0cl 28432 ax-hvaddid 28433 ax-hfvmul 28434 ax-hvmulid 28435 ax-hvdistr2 28438 ax-hvmul0 28439 ax-hfi 28508 ax-his1 28511 ax-his2 28512 ax-his3 28513 ax-his4 28514

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-2 11438 df-cj 14246 df-re 14247 df-im 14248 df-hvsub 28400 df-adjh 29280

This theorem is referenced by: nmopadjlei 29519

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