Theorem evl1fval 22469

Description: Value of the simple/same ring evaluation map. (Contributed by Mario Carneiro, 12-Jun-2015.)

Hypotheses

Ref	Expression
evl1fval.o	⊢ 𝑂 = (eval1‘𝑅)
evl1fval.q	⊢ 𝑄 = (1o eval 𝑅)
evl1fval.b	⊢ 𝐵 = (Base‘𝑅)

Assertion

Ref	Expression
evl1fval	⊢ 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑄   𝑥,𝑅

Allowed substitution hints:   𝑄(𝑦)   𝑅(𝑦)   𝑂(𝑥,𝑦)

Proof of Theorem evl1fval

Dummy variables 𝑖 𝑟 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evl1fval.o	. . 3 ⊢ 𝑂 = (eval1‘𝑅)
2		fvexd 6898	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
3		id 23	. . . . . . . . 9 ⊢ (𝑏 = (Base‘𝑟) → 𝑏 = (Base‘𝑟))
4		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5		evl1fval.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑅)
6	4, 5	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
7	3, 6	sylan9eqr 2820	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑏 = 𝐵)
8	7	oveq1d 7427	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑏 ↑m 1o) = (𝐵 ↑m 1o))
9	7, 8	oveq12d 7430	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑏 ↑m (𝑏 ↑m 1o)) = (𝐵 ↑m (𝐵 ↑m 1o)))
10	7	mpteq1d 5202	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))
11	10	coeq2d 5850	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦}))) = (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
12	9, 11	mpteq12dv 5199	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) = (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))))
13		simpl 487	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑟 = 𝑅)
14	13	oveq2d 7428	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (1o eval 𝑟) = (1o eval 𝑅))
15		evl1fval.q	. . . . . . 7 ⊢ 𝑄 = (1o eval 𝑅)
16	14, 15	eqtr4di 2816	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (1o eval 𝑟) = 𝑄)
17	12, 16	coeq12d 5852	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → ((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑟)) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
18	2, 17	csbied 3890	. . . 4 ⊢ (𝑟 = 𝑅 → ⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑟)) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
19		df-evl1 22457	. . . 4 ⊢ eval1 = (𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑟)))
20		ovex 7445	. . . . . 6 ⊢ (𝐵 ↑m (𝐵 ↑m 1o)) ∈ V
21	20	mptex 7223	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ V
22	15	ovexi 7446	. . . . 5 ⊢ 𝑄 ∈ V
23	21, 22	coex 7928	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄) ∈ V
24	18, 19, 23	fvmpt 6991	. . 3 ⊢ (𝑅 ∈ V → (eval1‘𝑅) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
25	1, 24	eqtrid 2810	. 2 ⊢ (𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
26		fvprc 6875	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (eval1‘𝑅) = ∅)
27	1, 26	eqtrid 2810	. . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅)
28		co02 6264	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ∅) = ∅
29	27, 28	eqtr4di 2816	. . 3 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ∅))
30		df-evl 22207	. . . . . . 7 ⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
31	30	reldmmpo 7546	. . . . . 6 ⊢ Rel dom eval
32	31	ovprc2 7452	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o eval 𝑅) = ∅)
33	15, 32	eqtrid 2810	. . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑄 = ∅)
34	33	coeq2d 5850	. . 3 ⊢ (¬ 𝑅 ∈ V → ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ∅))
35	29, 34	eqtr4d 2801	. 2 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
36	25, 35	pm2.61i 184	1 ⊢ 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ⦋csb 3854  ∅c0 4287  {csn 4590   ↦ cmpt 5193   × cxp 5661   ∘ ccom 5667  ‘cfv 6538  (class class class)co 7412  1_oc1o 8447   ↑_m cmap 8825  Basecbs 17270   evalSub ces 22204   eval cevl 22205  eval₁ce1 22455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-evl 22207  df-evl1 22457

This theorem is referenced by:  evl1val  22470  evl1fval1lem  22471  evl1rhm  22473  pf1rcl  22490