Theorem evl1fval 21694

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 6857	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
3		id 22	. . . . . . . . 9 ⊢ (𝑏 = (Base‘𝑟) → 𝑏 = (Base‘𝑟))
4		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5		evl1fval.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑅)
6	4, 5	eqtr4di 2794	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
7	3, 6	sylan9eqr 2798	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑏 = 𝐵)
8	7	oveq1d 7372	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑏 ↑m 1o) = (𝐵 ↑m 1o))
9	7, 8	oveq12d 7375	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑏 ↑m (𝑏 ↑m 1o)) = (𝐵 ↑m (𝐵 ↑m 1o)))
10	7	mpteq1d 5200	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))
11	10	coeq2d 5818	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦}))) = (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
12	9, 11	mpteq12dv 5196	. . . . . 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 483	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → 𝑟 = 𝑅)
14	13	oveq2d 7373	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (1o eval 𝑟) = (1o eval 𝑅))
15		evl1fval.q	. . . . . . 7 ⊢ 𝑄 = (1o eval 𝑅)
16	14, 15	eqtr4di 2794	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑏 = (Base‘𝑟)) → (1o eval 𝑟) = 𝑄)
17	12, 16	coeq12d 5820	. . . . 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 3893	. . . 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 21682	. . . 4 ⊢ eval1 = (𝑟 ∈ V ↦ ⦋(Base‘𝑟) / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑟)))
20		ovex 7390	. . . . . 6 ⊢ (𝐵 ↑m (𝐵 ↑m 1o)) ∈ V
21	20	mptex 7173	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ V
22	15	ovexi 7391	. . . . 5 ⊢ 𝑄 ∈ V
23	21, 22	coex 7867	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄) ∈ V
24	18, 19, 23	fvmpt 6948	. . 3 ⊢ (𝑅 ∈ V → (eval1‘𝑅) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
25	1, 24	eqtrid 2788	. 2 ⊢ (𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
26		fvprc 6834	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (eval1‘𝑅) = ∅)
27	1, 26	eqtrid 2788	. . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ∅)
28		co02 6212	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ∅) = ∅
29	27, 28	eqtr4di 2794	. . 3 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ∅))
30		df-evl 21483	. . . . . . 7 ⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
31	30	reldmmpo 7490	. . . . . 6 ⊢ Rel dom eval
32	31	ovprc2 7397	. . . . 5 ⊢ (¬ 𝑅 ∈ V → (1o eval 𝑅) = ∅)
33	15, 32	eqtrid 2788	. . . 4 ⊢ (¬ 𝑅 ∈ V → 𝑄 = ∅)
34	33	coeq2d 5818	. . 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 2779	. 2 ⊢ (¬ 𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
36	25, 35	pm2.61i 182	1 ⊢ 𝑂 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3445  ⦋csb 3855  ∅c0 4282  {csn 4586   ↦ cmpt 5188   × cxp 5631   ∘ ccom 5637  ‘cfv 6496  (class class class)co 7357  1_oc1o 8405   ↑_m cmap 8765  Basecbs 17083   evalSub ces 21480   eval cevl 21481  eval₁ce1 21680

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-evl 21483  df-evl1 21682

This theorem is referenced by:  evl1val  21695  evl1fval1lem  21696  evl1rhm  21698  pf1rcl  21715