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Theorem adjval 31713

Description: Value of the adjoint function for 𝑇 in the domain of adj_ℎ. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 24-Dec-2016.) (New usage is discouraged.)

**Assertion**
Ref	Expression
adjval	⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇) = (℩𝑢 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦)))

Distinct variable group: 𝑥,𝑢,𝑦,𝑇

Proof of Theorem adjval Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmadjop 31711	. . . . 5 ⊢ (𝑇 ∈ dom adj_ℎ → 𝑇: ℋ⟶ ℋ)
2	1	biantrurd 532	. . . 4 ⊢ (𝑇 ∈ dom adj_ℎ → ((𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ (𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦)))))
3		ax-hilex 30822	. . . . . 6 ⊢ ℋ ∈ V
4	3, 3	elmap 8890	. . . . 5 ⊢ (𝑢 ∈ ( ℋ ↑_m ℋ) ↔ 𝑢: ℋ⟶ ℋ)
5	4	anbi1i 623	. . . 4 ⊢ ((𝑢 ∈ ( ℋ ↑_m ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦)) ↔ (𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦)))
6		3anass 1093	. . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ (𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦))))
7	2, 5, 6	3bitr4g 314	. . 3 ⊢ (𝑇 ∈ dom adj_ℎ → ((𝑢 ∈ ( ℋ ↑_m ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦))))
8	7	iotabidv 6532	. 2 ⊢ (𝑇 ∈ dom adj_ℎ → (℩𝑢(𝑢 ∈ ( ℋ ↑_m ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦))) = (℩𝑢(𝑇: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦))))
9		df-riota 7376	. . 3 ⊢ (℩𝑢 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦)) = (℩𝑢(𝑢 ∈ ( ℋ ↑_m ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦)))
10	9	a1i 11	. 2 ⊢ (𝑇 ∈ dom adj_ℎ → (℩𝑢 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦)) = (℩𝑢(𝑢 ∈ ( ℋ ↑_m ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦))))
11		dfadj2 31708	. . 3 ⊢ adj_ℎ = {⟨𝑡, 𝑢⟩ ∣ (𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑡‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦))}
12		feq1 6703	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑡: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
13		fveq1 6896	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦))
14	13	oveq2d 7436	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑥 ·_ih (𝑡‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)))
15	14	eqeq1d 2730	. . . . 5 ⊢ (𝑡 = 𝑇 → ((𝑥 ·_ih (𝑡‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦) ↔ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦)))
16	15	2ralbidv 3215	. . . 4 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑡‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦)))
17	12, 16	3anbi13d 1435	. . 3 ⊢ (𝑡 = 𝑇 → ((𝑡: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑡‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦)) ↔ (𝑇: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦))))
18	11, 17	fvopab5 7038	. 2 ⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇) = (℩𝑢(𝑇: ℋ⟶ ℋ ∧ 𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦))))
19	8, 10, 18	3eqtr4rd 2779	1 ⊢ (𝑇 ∈ dom adj_ℎ → (adj_ℎ‘𝑇) = (℩𝑢 ∈ ( ℋ ↑_m ℋ)∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3058 dom cdm 5678 ℩cio 6498 ⟶wf 6544 ‘cfv 6548 ℩crio 7375 (class class class)co 7420 ↑_m cmap 8845 ℋchba 30742 ·_ih csp 30745 adj_ℎcado 30778

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-hilex 30822 ax-hfi 30902 ax-his1 30905

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-2 12306 df-cj 15079 df-re 15080 df-im 15081 df-adjh 31672

This theorem is referenced by: adjval2 31714 adjbdln 31906

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