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Theorem adjval 31976

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 31974	. . . . 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 31085	. . . . . 6 ⊢ ℋ ∈ V
4	3, 3	elmap 8812	. . . . 5 ⊢ (𝑢 ∈ ( ℋ ↑_m ℋ) ↔ 𝑢: ℋ⟶ ℋ)
5	4	anbi1i 625	. . . 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 1095	. . . 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 6476	. 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 7317	. . 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 31971	. . 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 6640	. . . 4 ⊢ (𝑡 = 𝑇 → (𝑡: ℋ⟶ ℋ ↔ 𝑇: ℋ⟶ ℋ))
13		fveq1 6833	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦))
14	13	oveq2d 7376	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑥 ·_ih (𝑡‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)))
15	14	eqeq1d 2739	. . . . 5 ⊢ (𝑡 = 𝑇 → ((𝑥 ·_ih (𝑡‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦) ↔ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦)))
16	15	2ralbidv 3202	. . . 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 1441	. . 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 6975	. 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 2783	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 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 dom cdm 5624 ℩cio 6446 ⟶wf 6488 ‘cfv 6492 ℩crio 7316 (class class class)co 7360 ↑_m cmap 8766 ℋchba 31005 ·_ih csp 31008 adj_ℎcado 31041

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-hilex 31085 ax-hfi 31165 ax-his1 31168

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-2nd 7936 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-er 8636 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-cj 15052 df-re 15053 df-im 15054 df-adjh 31935

This theorem is referenced by: adjval2 31977 adjbdln 32169

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